Function compositionFrom Wikipedia, the free encyclopedia

Contents

1 Function composition 11.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.3 Composition monoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.4 Functional powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.5 Alternative notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.6 Composition operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.7 In programming languages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.8 Multivariate functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.9 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.10 Typography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.11 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61.12 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.13 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.14 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Inverse function 82.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

2.1.1 Example: squaring and square root functions . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.2 Inverses in higher mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.1.3 Inverses and composition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.1.4 Note on notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

2.2 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.1 Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.2 Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2.3 Self-inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

2.3 Inverses in calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.1 Formula for the inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102.3.2 Graph of the inverse . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.3.3 Inverses and derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

2.4 Real-world examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5 Generalizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.5.1 Partial inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

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2.5.2 Left and right inverses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.5.3 Preimages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142.8 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.9 Further reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.10 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3 Inverse trigonometric functions 223.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.1.1 Etymology of the arc- prefix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2 Basic properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3.2.1 Principal values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.2.2 Relationships between trigonometric functions and inverse trigonometric functions . . . . . 233.2.3 Relationships among the inverse trigonometric functions . . . . . . . . . . . . . . . . . . . 233.2.4 Arctangent addition formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.3 In calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.3.1 Derivatives of inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . . . . . 243.3.2 Expression as definite integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.3 Infinite series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3.4 Indefinite integrals of inverse trigonometric functions . . . . . . . . . . . . . . . . . . . . 26

3.4 Extension to complex plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4.1 Logarithmic forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28

3.5 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5.1 General solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303.5.2 In computer science and engineering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3.6 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.7 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.8 External links . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

4 Multiplicative inverse 364.1 Examples and counterexamples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.2 Complex numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374.3 Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384.4 Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.5 Reciprocals of irrational numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.6 Further remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394.7 Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.8 See also . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404.9 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.10 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 414.11 Text and image sources, contributors, and licenses . . . . . . . . . . . . . . . . . . . . . . . . . . 42

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4.11.1 Text . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 424.11.2 Images . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 434.11.3 Content license . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

Chapter 1

Function composition

In mathematics, function composition is the pointwise application of one function to the result of another to producea third function. For instance, the functions f : X → Y and g : Y → Z can be composed to yield a function whichmaps x in X to g(f(x)) in Z. Intuitively, if z is a function of y, and y is a function of x, then z is a function of x. Theresulting composite function is denoted g ∘ f : X→ Z, defined by (g ∘ f )(x) = g(f(x)) for all x in X.[note 1] The notationg ∘ f is read as "g circle f ", or "g round f ", or "g composed with f ", "g after f ", "g following f ", or "g of f", or "gon f ". Intuitively, composing two functions is a chaining process in which the output of the first function becomesthe input of the second function.The composition of functions is a special case of the composition of relations, so all properties of the latter are trueof composition of functions.[1] The composition of function has some additional properties.

1.1 Examples

X Y Zf g

a

bc

d

1

2

3

4

@

#

!!

g ∘ f , the composition of f and g. For example, (g ∘ f )(c) = #.

• Composition of functions on a finite set: If f = {(1,3), (2,1), (3,4), (4,6)}, and g = {(1,5), (2,3), (3,4), (4,1),(5,3), (6,2)}, then g ∘ f = {(1,4), (2,5), (3,1), (4,2)}.

• Composition of functions on an infinite set: If f: ℝ → ℝ (where ℝ is the set of all real numbers) is given byf(x) = 2x + 4 and g: ℝ → ℝ is given by g(x) = x3, then:

(f ∘ g)(x) = f(g(x)) = f(x3) = 2x3 + 4, and

1

2 CHAPTER 1. FUNCTION COMPOSITION

(g ∘ f)(x) = g(f(x)) = g(2x + 4) = (2x + 4)3.

• If an airplane’s elevation at time t is given by the function h(t), and the oxygen concentration at elevation x isgiven by the function c(x), then (c ∘ h)(t) describes the oxygen concentration around the plane at time t.

1.2 Properties

The composition of functions is always associative—a property inherited from the composition of relations.[1] Thatis, if f, g, and h are three functions with suitably chosen domains and codomains, then f ∘ (g ∘ h) = (f ∘ g) ∘ h, wherethe parentheses serve to indicate that composition is to be performed first for the parenthesized functions. Since thereis no distinction between the choices of placement of parentheses, they may be left off without causing any ambiguity.In a strict sense, the composition g ∘ f can be built only if f 's codomain equals g's domain; in a wider sense it issufficient that the former is a subset of the latter.[note 2] Moreover, it is often convenient to tacitly restrict f 's domainsuch that f produces only values in g's domain; for example, the composition g ∘ f of the functions f : ℝ → (−∞,+9]defined by f(x) = 9 − x2 and g : [[interval (mathematics)#Infinite endpoints|[0,+∞)]] → ℝ defined by g(x) = √x canbe defined on the interval [−3,+3].The functions g and f are said to commute with each other if g ∘ f = f ∘ g. Commutativity is a special property,attained only by particular functions, and often in special circumstances. For example, |x| + 3 = |x + 3| only when x ≥0. The picture shows another example.The composition of one-to-one functions is always one-to-one. Similarly, the composition of two onto functions isalways onto. It follows that composition of two bijections is also a bijection. The inverse function of a composition(assumed invertible) has the property that (f ∘ g)−1 = ( g−1 ∘ f −1).[2]

Derivatives of compositions involving differentiable functions can be found using the chain rule. Higher derivativesof such functions are given by Faà di Bruno’s formula.

1.3 Composition monoids

Main article: Transformation monoid

Suppose one has two (or more) functions f: X → X, g: X → X having the same domain and codomain; these areoften called transformations. Then one can form chains of transformations composed together, such as f ∘ f ∘ g ∘f. Such chains have the algebraic structure of a monoid, called a transformation monoid or (much more seldom)composition monoid. In general, transformation monoids can have remarkably complicated structure. One particularnotable example is the de Rham curve. The set of all functions f: X→X is called the full transformation semigroup[3]or symmetric semigroup[4] on X. (One can actually define two semigroups depending how one defines the semigroupoperation as the left or right composition of functions.[5])If the transformation are bijective (and thus invertible), then the set of all possible combinations of these functionsforms a transformation group; and one says that the group is generated by these functions. A fundamental resultin group theory, Cayley’s theorem, essentially says that any group is in fact just a group of permutations (up toisomorphism).[6]

The set of all bijective functions f: X → X (called permutations) forms a group with respect to the compositionoperator. This is the symmetric group, also sometimes called the composition group.In the symmetric semigroup (of all transformations) one also finds a weaker, non-unique notion of inverse (called apseudoinverse) because the symmetric semigroup is a regular semigroup.[7]

1.4 Functional powers

Main article: Iterated function

If Y ⊆ X, then f: X→Y may compose with itself; this is sometimes denoted as f 2. That is:

1.4. FUNCTIONAL POWERS 3

x

y

f (| x |)

| f (x ) |

f (x )

Compositions of two real functions, absolute value and a cubic function, in different orders show a non-commutativity of the com-position.

(f ∘ f)(x) = f(f(x)) = f 2(x)

(f ∘ f ∘ f)(x) = f(f(f(x))) = f 3(x)

More generally, for any natural number n ≥ 2, the nth functional power can be defined inductively by f n = f ∘ f n−1

= f n−1 ∘ f. Repeated composition of such a function with itself is called iterated function.

• By convention, f 0 is defined as the identity map on f 's domain, idX.

4 CHAPTER 1. FUNCTION COMPOSITION

135ASB

S

V

W

U

E

C

A

F

B

T

The similarity that transforms triangle EFA into triangle ATB is the composition of a homothety H and a rotation R, of which thecommon centre is S. For example, the image of A under the rotation R is U, which may be written R (A) = U. And H(U) = Bmeans that the mapping H transforms U into B. Thus H(R (A)) = (H ∘ R )(A) = B.

• If even Y = X and f: X→ X admits an inverse function f −1, negative functional powers f −n are defined for n> 0 as the opposite power of the inverse function: f −n = (f −1)n.

Note: If f takes its values in a ring (in particular for real or complex-valued f ), there is a risk of confusion, as f n

could also stand for the n-fold product of f, e.g. f 2(x) = f(x) · f(x). For trigonometric functions, usually the latteris meant, at least for positive exponents. For example, in trigonometry, this superscript notation represents standardexponentiation when used with trigonometric functions: sin2(x) = sin(x) · sin(x). However, for negative exponents(especially −1), it nevertheless usually refers to the inverse function, e.g., tan−1 = arctan (≠ 1/tan).In some cases, when, for a given function f, the equation g ∘ g = f has a unique solution g, that function can be definedas the functional square root of f, then written as g = f 1/2.More generally, when gn = f has a unique solution for some natural number n > 0, then f m/n can be defined as gm.Under additional restrictions, this idea can be generalized so that the iteration count becomes a continuous parameter;in this case, such a system is called a flow, specified through solutions of Schröder’s equation. Iterated functions andflows occur naturally in the study of fractals and dynamical systems.

1.5. ALTERNATIVE NOTATIONS 5

1.5 Alternative notations

Many mathematicians, particularly in group theory, omit the composition symbol, writing gf for g ∘ f.[8]

In the mid-20th century, some mathematicians decided that writing "g ∘ f " to mean “first apply f, then apply g" wastoo confusing and decided to change notations. They write "xf " for "f(x)" and "(xf)g" for "g(f(x))".[9] This can bemore natural and seem simpler than writing functions on the left in some areas – in linear algebra, for instance, when xis a row vector and f and g denote matrices and the composition is by matrix multiplication. This alternative notationis called postfix notation. The order is important because matrix multiplication is non-commutative. Successivetransformations applying and composing to the right agrees with the left-to-right reading sequence.Mathematicians who use postfix notation may write "fg", meaning first apply f and then apply g, in keeping withthe order the symbols occur in postfix notation, thus making the notation "fg" ambiguous. Computer scientists maywrite "f ; g" for this,[10] thereby disambiguating the order of composition. To distinguish the left composition operatorfrom a text semicolon, in the Z notation the character is used for left relation composition.[11] Since all functions arebinary relations, it is correct to use the [fat] semicolon for function composition as well (see the article on compositionof relations for further details on this notation).

1.6 Composition operator

Main article: Composition operator

Given a function g, the composition operator Cg is defined as that operator which maps functions to functions as

Cgf = f ◦ g.

Composition operators are studied in the field of operator theory.

1.7 In programming languages

Main article: Function composition (computer science)

Function composition appears in one form or another in numerous programming languages.

1.8 Multivariate functions

Partial composition is possible for multivariate functions. The function resulting when some argument xi of thefunction f is replaced by the function g is called a composition of f and g in some computer engineering contexts,and is denoted f |xi ₌ g

f |xi=g = f(x1, . . . , xi−1, g(x1, x2, . . . , xn), xi+1, . . . , xn).

When g is a simple constant b, composition degenerates into a (partial) valuation, whose result is also known asrestriction or co-factor.[12]

f |xi=b = f(x1, . . . , xi−1, b, xi+1, . . . , xn).

In general, the composition of multivariate functions may involve several other functions as arguments, as in thedefinition of primitive recursive function. Given f, a n-ary function, and n m-ary functions g1, ..., gn, the compositionof f with g1, ..., gn, is the m-ary function

6 CHAPTER 1. FUNCTION COMPOSITION

h(x1, . . . , xm) = f(g1(x1, . . . , xm), . . . , gn(x1, . . . , xm))

This is sometimes called the generalized composite of f with g1, ..., gn.[13] The partial composition in only one argu-ment mentioned previously can be instantiated from this more general scheme by setting all argument functions exceptone to be suitably chosen projection functions. Note also that g1, ..., gn can be seen as a single vector/tuple-valuedfunction in this generalized scheme, in which case this is precisely the standard definition of function composition.[14]

A set of finitary operations on some base set X is called a clone if it contains all projections and is closed undergeneralized composition. Note that a clone generally contains operations of various arities.[13] The notion of com-mutation also finds an interesting generalization in the multivariate case; a function f of arity n is said to commutewith a function g of arity m if f is a homomorphism preserving g, and vice versa i.e.:[15]

f(g(a11, . . . , a1m), . . . , g(an1, . . . , anm)) = g(f(a11, . . . , an1), . . . , g(a1m, . . . , amn))

A unary operation always commutes with itself, but this is not necessarily the case for a binary (or higher arity)operation. A binary (or higher arity) operation that commutes with itself is called medial or entropic.[15]

1.9 Generalizations

Composition can be generalized to arbitrary binary relations. If R⊆X × Y and S ⊆ Y × Z are two binary relations, thentheir composition S∘R is the relation defined as {(x, z) ∈ X × Z : ∃y ∈ Y. (x, y) ∈ R ∧ (y, z) ∈ S}. Considering a functionas a special case of a binary relation (namely functional relations), function composition satisfies the definition forrelation composition.The composition is defined in the same way for partial functions and Cayley’s theorem has its analogue calledWagner-Preston theorem.[16]

The category of sets with functions as morphisms is the prototypical category. The axioms of a category are in factinspired from the properties (and also the definition) of function composition.[17] The structures given by compo-sition are axiomatized and generalized in category theory with the concept of morphism as the category-theoreticalreplacement of functions. The order inversion in the formula (f ∘ g)−1 = (g−1 ∘ f −1) applies for groups in general andfor the inverse relation; each of these is a dagger category.

1.10 Typography

The composition symbol ∘ is encoded as U+2218 ∘ ring operator (HTML ∘); see the Degree symbol articlefor similar-appearing Unicode characters. In TeX, it is written \circ.

1.11 See also

• Combinatory logic

• Function composition (computer science)

• Functional decomposition

• Iterated function

• Flow (mathematics)

• Higher-order function

• Cobweb plot – a graphical technique for functional composition

• Lambda calculus

1.12. NOTES 7

• Functional square root

• Composition ring, a formal axiomatization of the composition operation

• Function of random variable, distribution of a function of a random variable

1.12 Notes[1] Some authors use f ∘ g : X→ Z, defined by (f ∘ g )(x) = g(f(x)) instead.

[2] The strict sense is used, e.g., in category theory, where a subset relation is modelled explicitly by an inclusion function.

1.13 References[1] Daniel J. Velleman (2006). How to Prove It: A Structured Approach. Cambridge University Press. p. 232. ISBN 978-1-

139-45097-3.

[2] Nancy Rodgers (2000). Learning to Reason: An Introduction to Logic, Sets, and Relations. John Wiley & Sons. pp.359–362. ISBN 978-0-471-37122-9.

[3] Christopher Hollings (2014). Mathematics across the Iron Curtain: A History of the Algebraic Theory of Semigroups.American Mathematical Society. p. 334. ISBN 978-1-4704-1493-1.

[4] Pierre A. Grillet (1995). Semigroups: An Introduction to the Structure Theory. CRC Press. p. 2. ISBN 978-0-8247-9662-4.

[5] Pál Dömösi; Chrystopher L. Nehaniv (2005). Algebraic Theory of Automata Networks: A Introduction. SIAM. p. 8. ISBN978-0-89871-569-9.

[6] Nathan Carter (9 April 2009). Visual Group Theory. MAA. p. 95. ISBN 978-0-88385-757-1.

[7] Olexandr Ganyushkin; Volodymyr Mazorchuk (2008). Classical Finite Transformation Semigroups: An Introduction.Springer Science & Business Media. p. 24. ISBN 978-1-84800-281-4.

[8] Oleg A. Ivanov (1 January 2009). Making Mathematics Come to Life: A Guide for Teachers and Students. AmericanMathematical Soc. pp. 217–. ISBN 978-0-8218-4808-1.

[9] Jean Gallier (2011). Discrete Mathematics. Springer. p. 118. ISBN 978-1-4419-8047-2.

[10] Michael Barr; Charles Wells (1998). Category Theory for Computing Science (PDF). p. 6. This is the updated and freeversion of book originally published by Prentice Hall in 1990 as ISBN 978-0-13-120486-7.

[11] ISO/IEC 13568:2002(E), p. 23

[12] Bryant, R.E. (August 1986). “LogicMinimization Algorithms for VLSI Synthesis” (PDF). IEEE Transactions on ComputersC–35 (8): 677–691. doi:10.1109/tc.1986.1676819.

[13] Clifford Bergman (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 79–80. ISBN 978-1-4398-5129-6.

[14] George Tourlakis (2012). Theory of Computation. John Wiley & Sons. p. 100. ISBN 978-1-118-31533-0.

[15] Clifford Bergman (2011). Universal Algebra: Fundamentals and Selected Topics. CRC Press. pp. 90–91. ISBN 978-1-4398-5129-6.

[16] S. Lipscomb, “Symmetric Inverse Semigroups”, AMSMathematical Surveys andMonographs (1997), ISBN 0-8218-0627-0, p. xv

[17] Peter Hilton; Yel-Chiang Wu (1989). A Course in Modern Algebra. John Wiley & Sons. p. 65. ISBN 978-0-471-50405-4.

1.14 External links• Hazewinkel, Michiel, ed. (2001), “Composite function”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• "Composition of Functions" by Bruce Atwood, the Wolfram Demonstrations Project, 2007.

Chapter 2

Inverse function

In mathematics, an inverse function is a function that “reverses” another function. That is, if f is a function mappingx to y, then the inverse function of f maps y back to x.[1]

2.1 Definitions

See also: Inverse element

Let f be a function whose domain is the set X, and whose image (range) is the set Y. Then f is invertible if there existsa function g with domain Y and image X, with the property:

f(x) = y ⇔ g(y) = x.

If f is invertible, the function g is unique, which means that there is exactly one function g satisfying this property (nomore, no less). That function g is then called the inverse of f, and is usually denoted as f −1.Stated otherwise, a function is invertible if and only if its inverse relation is a function on the range Y, in which casethe inverse relation is the inverse function.[2]

Not all functions have an inverse. For this rule to be applicable, each element y ∈ Y must correspond to no more thanone x ∈ X; a function f with this property is called one-to-one or an injection. If f and f −1 are functions on X andY respectively, then both are bijections. The inverse of an injection that is not a bijection is a partial function, thatmeans for some y ∈ Y it is undefined.

2.1.1 Example: squaring and square root functions

The function f(x) = x2 may or may not be invertible, depending on what kinds of numbers are being considered (the“domain”).If the domain is the real numbers, then each possible result y (except 0) corresponds to two different starting pointsin X – one positive and one negative, and so this function is not invertible: as it is impossible to deduce an input fromits output. Such a function is called non-injective or information-losing.If the domain of the function is restricted to the nonnegative reals then the function is injective and invertible.

2.1.2 Inverses in higher mathematics

The definition given above is commonly adopted in set theory and calculus. In higher mathematics, the notation

f : X → Y

8

2.2. PROPERTIES 9

means “f is a function mapping elements of a set X to elements of a set Y ". The source, X, is called the domainof f, and the target, Y, is called the codomain. The codomain contains the range of f as a subset, and is part of thedefinition of f.[3]

When using codomains, the inverse of a function f: X → Y is required to have domain Y and codomain X. For theinverse to be defined on all of Y, every element of Y must lie in the range of the function f. A function with thisproperty is called onto or surjective. Thus, a function with a codomain is invertible if and only if it is both injective(one-to-one) and surjective (onto). Such a function is called a one-to-one correspondence or a bijection, and has theproperty that every element y ∈ Y corresponds to exactly one element x ∈ X.

2.1.3 Inverses and composition

If f is an invertible function with domain X and range Y, then

f−1 ( f(x) ) = x , for every x ∈ X.

Using the composition of functions we can rewrite this statement as follows:

f−1 ◦ f = idX ,

where idX is the identity function on the set X; that is, the function that leaves its argument unchanged. In categorytheory, this statement is used as the definition of an inverse morphism.Considering function composition helps to understand the notation f −1. Repeatedly composing a function with itselfis called iteration.If f is applied n times, starting with the value x, then this is written as f n(x); so f 2(x) = f (f (x)),etc. Since f −1(f (x)) = x, composing f −1 and f n yields f n−1, “undoing” the effect of one application of f.

2.1.4 Note on notation

Whereas the notation f −1(x) might be misunderstood, f(x)−1 certainly denotes the multiplicative inverse of f(x) andhas nothing to do with inversion of f.The expression sin−1 x does not represent the multiplicative inverse to sin x,[4] but the inverse of the sine functionapplied to x (actually a partial inverse; see below). To avoid confusion, an inverse trigonometric function is oftenindicated by the prefix "arc" (for Latin arcus). For instance, the inverse of the sine function is typically called thearcsine function, written as arcsin. Similarly, the inverse of a hyperbolic function is indicated by the prefix "ar" (forLatin area).

2.2 Properties

2.2.1 Uniqueness

If an inverse function exists for a given function f, it is unique: it must be the inverse relation.

2.2.2 Symmetry

There is a symmetry between a function and its inverse. Specifically, if f is an invertible function with domain X andrange Y, then its inverse f −1 has domain Y and range X, and the inverse of f −1 is the original function f. In symbols,for functions f:X→Y and g:Y→X,

g ◦ f = idX ⇒ f ◦ g = idY .

This follows from the connection between function inverse and relation inverse, because inversion of relations is aninvolution.

10 CHAPTER 2. INVERSE FUNCTION

This statement is an obvious consequence of the deduction that for f to be invertible it must be injective (first definitionof the inverse) or bijective (second definition). The property of involutive symmetry can be concisely expressed bythe following formula:

(f−1

)−1= f.

The inverse of a composition of functions is given by the formula

(g ◦ f)−1 = f−1 ◦ g−1

Notice that the order of g and f have been reversed; to undo f followed by g, we must first undo g and then undo f.For example, let f(x) = 3x and let g(x) = x + 5. Then the composition g ∘ f is the function that first multiplies by threeand then adds five:

(g ◦ f)(x) = 3x+ 5

To reverse this process, we must first subtract five, and then divide by three:

(g ◦ f)−1(y) = 13 (y − 5)

This is the composition (f −1 ∘ g −1)(y).

2.2.3 Self-inverses

If X is a set, then the identity function on X is its own inverse:

idX−1 = idX

More generally, a function f : X → X is equal to its own inverse if and only if the composition f ∘ f is equal to idX.Such a function is called an involution.

2.3 Inverses in calculus

Single-variable calculus is primarily concerned with functions that map real numbers to real numbers. Such functionsare often defined through formulas, such as:

f(x) = (2x+ 8)3.

A function f from the real numbers to the real numbers possesses an inverse as long as it is one-to-one, i.e. as longas the graph of y = f(x) has, for each possible y value only one corresponding x value, and thus passes the horizontalline test.The following table shows several standard functions and their inverses:

2.3.1 Formula for the inverse

One approach to finding a formula for f −1, if it exists, is to solve the equation y = f(x) for x. For example, if f is thefunction

2.3. INVERSES IN CALCULUS 11

f(x) = (2x+ 8)3

then we must solve the equation y = (2x + 8)3 for x:

y = (2x+ 8)3

3√y = 2x+ 8

3√y − 8 = 2x

3√y − 8

2= x.

Thus the inverse function f −1 is given by the formula

f−1(y) =3√y − 8

2.

Sometimes the inverse of a function cannot be expressed by a formula with a finite number of terms. For example,if f is the function

f(x) = x− sinx,

then f is one-to-one, and therefore possesses an inverse function f −1. The formula for this inverse has an infinitenumber of terms:

f−1(y) =∞∑

n=1

yn3

n!limθ→0

(dn−1

dθ n−1

(θ

3√θ − sin(θ)

n))

2.3.2 Graph of the inverse

If f is invertible, then the graph of the function

y = f−1(x)

is the same as the graph of the equation

x = f(y).

This is identical to the equation y = f(x) that defines the graph of f, except that the roles of x and y have been reversed.Thus the graph of f −1 can be obtained from the graph of f by switching the positions of the x and y axes. This isequivalent to reflecting the graph across the line y = x.

2.3.3 Inverses and derivatives

A continuous function f is one-to-one (and hence invertible) if and only if it is either strictly increasing or decreasing(with no local maxima or minima). For example, the function

f(x) = x3 + x

12 CHAPTER 2. INVERSE FUNCTION

is invertible, since the derivative f′ (x) = 3x2 + 1 is always positive.If the function f is differentiable, then the inverse f −1 will be differentiable as long as f′ (x) ≠ 0. The derivative of theinverse is given by the inverse function theorem:

(f−1

)′(y) =

1

f ′ (f−1(y)).

If we set y = f(x), then the formula above can be written

dx

dy=

1

dy/dx.

This result follows from the chain rule (see the article on inverse functions and differentiation).The inverse function theorem can be generalized to functions of several variables. Specifically, a differentiablemultivariable function f : Rn → Rn is invertible in a neighborhood of a point p as long as the Jacobian matrix of f atp is invertible. In this case, the Jacobian of f −1 at f(p) is the matrix inverse of the Jacobian of f at p.

2.4 Real-world examples

1. Let f be the function that converts a temperature in degrees Celsius to a temperature in degrees Fahrenheit:

F = f(C) = 95C + 32;

then its inverse function converts degrees Fahrenheit to degrees Celsius:

C = f−1(F ) = 59 (F − 32),

since

f−1 ( f(C) ) = f−1(

95C + 32

)= 5

9

((95C + 32

)− 32

)= C every for ,C.

2. Suppose f assigns each child in a family its birth year. An inverse function would output which child was born ina given year. However, if the family has twins (or triplets) then the output cannot be known when the input is thecommon birth year. As well, if a year is given in which no child was born then a child cannot be named. But if eachchild was born in a separate year, and if we restrict attention to the three years in which a child was born, then we dohave an inverse function. For example,

f(Allan) = 2005, f(Brad) = 2007, f(Cary) = 2001

f−1(2005) = Allan, f−1(2007) = Brad, f−1(2001) = Cary

3. Let R be the function that leads to an x percentage rise of some quantity, and F be the function producing an xpercentage fall. Applied to $100 with x = 10%, we find that applying the first function followed by the second doesnot restore the original value of $100, demonstrating the fact that, despite appearances, these two functions are notinverses of each other.

2.5 Generalizations

2.5. GENERALIZATIONS 13

2.5.1 Partial inverses

Even if a function f is not one-to-one, it may be possible to define a partial inverse of f by restricting the domain.For example, the function

f(x) = x2

is not one-to-one, since x2 = (−x)2. However, the function becomes one-to-one if we restrict to the domain x ≥ 0, inwhich case

f−1(y) =√y.

(If we instead restrict to the domain x ≤ 0, then the inverse is the negative of the square root of y.) Alternatively,there is no need to restrict the domain if we are content with the inverse being a multivalued function:

f−1(y) = ±√y.

Sometimes this multivalued inverse is called the full inverse of f, and the portions (such as √x and −√x) are calledbranches. The most important branch of a multivalued function (e.g. the positive square root) is called the principalbranch, and its value at y is called the principal value of f −1(y).For a continuous function on the real line, one branch is required between each pair of local extrema. For example,the inverse of a cubic function with a local maximum and a local minimum has three branches (see the picture to theright).These considerations are particularly important for defining the inverses of trigonometric functions. For example, thesine function is not one-to-one, since

sin(x+ 2π) = sin(x)

for every real x (and more generally sin(x + 2πn) = sin(x) for every integer n). However, the sine is one-to-one onthe interval [−π/2, π/2], and the corresponding partial inverse is called the arcsine. This is considered the principalbranch of the inverse sine, so the principal value of the inverse sine is always between −π/2 and π/2. The followingtable describes the principal branch of each inverse trigonometric function:

2.5.2 Left and right inverses

If f: X→ Y, a left inverse for f (or retraction of f) is a function g: Y → X such that

g ◦ f = idX .

That is, the function g satisfies the rule

If f(x) = y , then g(y) = x.

Thus, g must equal the inverse of f on the image of f, but may take any values for elements of Y not in the image. Afunction f with a left inverse is necessarily injective. In classical mathematics, every injective function f necessarilyhas a left inverse; however, this may fail in constructive mathematics. For instance, a left inverse of the inclusion{0,1} → R of the two-element set in the reals violates indecomposability by giving a retraction of the real line to theset {0,1} .A right inverse for f (or section of f) is a function h: Y → X such that

14 CHAPTER 2. INVERSE FUNCTION

f ◦ h = idY .

That is, the function h satisfies the rule

If h(y) = x , then f(x) = y.

Thus, h(y) may be any of the elements of X that map to y under f. A function f has a right inverse if and only if it issurjective (though constructing such an inverse in general requires the axiom of choice).An inverse which is both a left and right inverse must be unique. Likewise, if g is a left inverse for f, then g mayor may not be a right inverse for f; and if g is a right inverse for f, then g is not necessarily a left inverse for f. Forexample let f: R→ [0, ∞) denote the squaring map, such that f(x) = x2 for all x in R, and let g: [0, ∞) → R denotethe square root map, such that g(x) = √x for all x ≥ 0. Then f(g(x)) = x for all x in [0, ∞); that is, g is a right inverseto f. However, g is not a left inverse to f, since, e.g., g(f(−1)) = 1 ≠ −1.

2.5.3 Preimages

If f: X→ Y is any function (not necessarily invertible), the preimage (or inverse image) of an element y ∈ Y is theset of all elements of X that map to y:

f−1({y}) = {x ∈ X : f(x) = y} .

The preimage of y can be thought of as the image of y under the (multivalued) full inverse of the function f.Similarly, if S is any subset of Y, the preimage of S is the set of all elements of X that map to S:

f−1(S) = {x ∈ X : f(x) ∈ S} .

For example, take a function f: R→ R, where f: x↦ x2. This function is not invertible for reasons discussed above.Yet preimages may be defined for subsets of the codomain:

f−1({1, 4, 9, 16}) = {−4,−3,−2,−1, 1, 2, 3, 4}

The preimage of a single element y ∈ Y – a singleton set {y} – is sometimes called the fiber of y. When Y is the setof real numbers, it is common to refer to f −1({y}) as a level set.

2.6 See also• Inverse function theorem, gives sufficient conditions for a function to be invertible in a neighborhood of a pointin its domain and gives a formula for the derivative of the inverse function

• Inverse functions and differentiation

• Inverse relation

• Lagrange inversion theorem, gives the Taylor series expansion of the inverse function of an analytic function

2.7 Notes[1] Keisler, H. Jerome. “Differentiation” (PDF). Retrieved 2015-01-24. § 2.4

[2] Smith, Eggen & St. Andre 2006, p. 202, Theorem 4.9

[3] Smith, Eggen & St. Andre 2006, p. 179

[4] Thomas 1972, pp. 304-309

2.8. REFERENCES 15

2.8 References• Smith, Douglas; Eggen, Maurice; St. Andre, Richard (2006), A Transition to Advanced Mathematics (6th ed.),Thompson Brooks/Cole, ISBN 978-0-534-39900-9

• Thomas, Jr., George B. (1972), Calculus and Analytic Geometry Part 1: Functions of One Variable and AnalyticGeometry (Alternate ed.), Addison-Wesley

2.9 Further reading• Spivak, Michael (1994), Calculus (3rd ed.), Publish or Perish, ISBN 0-914098-89-6

• Stewart, James (2002), Calculus (5th ed.), Brooks Cole, ISBN 978-0-534-39339-7

2.10 External links• Hazewinkel, Michiel, ed. (2001), “Inverse function”, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4

• Wikibook: Functions

• Wolfram Mathworld: Inverse Function

16 CHAPTER 2. INVERSE FUNCTION

A function f and its inverse f −1. Because f maps a to 3, the inverse f −1 maps 3 back to a.

2.10. EXTERNAL LINKS 17

If f maps X to Y, then f −1 maps Y back to X.

The inverse of g ∘ f is f −1 ∘ g −1.

18 CHAPTER 2. INVERSE FUNCTION

The graphs of y = f(x) and y = f −1(x). The dotted line is y = x.

2.10. EXTERNAL LINKS 19

y

x

y = x2

y = √x̅

x = y

The square root of x is a partial inverse to f(x) = x2.

20 CHAPTER 2. INVERSE FUNCTION

The inverse of this cubic function has three branches.

2.10. EXTERNAL LINKS 21

The arcsine is a partial inverse of the sine function.

Chapter 3

Inverse trigonometric functions

In mathematics, the inverse trigonometric functions (occasionally called cyclometric functions[1]) are the inversefunctions of the trigonometric functions (with suitably restricted domains). Specifically, they are the inverses of thesine, cosine, tangent, cotangent, secant, and cosecant functions. They are used to obtain an angle from any of theangle’s trigonometric ratios. Inverse trigonometric functions are widely used in engineering, navigation, physics, andgeometry.

3.1 Notation

There are many notations used for the inverse trigonometric functions. The notations sin−1 (x), cos−1 (x), tan−1(x), etc. are often used, but this convention logically conflicts with the common semantics for expressions like sin2(x), which refer to numeric power rather than function composition, and therefore may result in confusion betweenmultiplicative inverse and compositional inverse. The confusion is somewhat ameliorated by the fact that each of thereciprocal trigonometric functions has its own name—for example, (cos(x))−1=sec(x). Another convention used bysome authors[2] is to use a majuscule (capital/upper-case) first letter along with a −1 superscript, e.g., Sin−1 (x), Cos−1(x), etc., which avoids confusing them with the multiplicative inverse, which should be represented by sin−1 (x), cos−1(x), etc. Yet another convention is to use an arc- prefix, so that the confusion with the −1 superscript is resolvedcompletely, e.g., arcsin (x), arccos (x), etc. This convention is used throughout the article. In computer programminglanguages (also MS Office Excel) the inverse trigonometric functions are usually called asin, acos, atan.According to Cajori,[3] the notation sin−1 (x) was introduced by John Herschel in 1813.[4]

3.1.1 Etymology of the arc- prefix

When measuring in radians, an angle of θ radians will correspond to an arc whose length is rθ, where r is the radiusof the circle. Thus, in the unit circle, “the arc whose cosine is x” is the same as “the angle whose cosine is x”, becausethe length of the arc of the circle in radii is the same as the measurement of the angle in radians.[5]

3.2 Basic properties

3.2.1 Principal values

Since none of the six trigonometric functions are one-to-one, they are restricted in order to have inverse functions.Therefore the ranges of the inverse functions are proper subsets of the domains of the original functionsFor example, using function in the sense of multivalued functions, just as the square root function y = √x could bedefined from y2 = x, the function y = arcsin(x) is defined so that sin(y) = x. There are multiple numbers y such thatsin(y) = x; for example, sin(0) = 0, but also sin(π) = 0, sin(2π) = 0, etc. When only one value is desired, the functionmay be restricted to its principal branch. With this restriction, for each x in the domain the expression arcsin(x)

22

3.2. BASIC PROPERTIES 23

will evaluate only to a single value, called its principal value. These properties apply to all the inverse trigonometricfunctions.The principal inverses are listed in the following table.(Note: Some authors define the range of arcsecant to be ( 0 ≤ y < π/2 or π ≤ y < 3π/2 ), because the tangentfunction is nonnegative on this domain. This makes some computations more consistent. For example using thisrange, tan(arcsec(x))=√x2−1, whereas with the range ( 0 ≤ y < π/2 or π/2 < y ≤ π ), we would have to writetan(arcsec(x))=±√x2−1, since tangent is nonnegative on 0 ≤ y < π/2 but nonpositive on π/2 < y ≤ π. For a similarreason, the same authors define the range of arccosecant to be ( -π < y ≤ -π/2 or 0 < y ≤ π/2 ).)If x is allowed to be a complex number, then the range of y applies only to its real part.

3.2.2 Relationships between trigonometric functions and inverse trigonometric functions

Trigonometric functions of inverse trigonometric functions are tabulated below. A quick way to derive them is byconsidering the geometry of a right-angled triangle, with one side of length 1, and another side of length x (any realnumber between 0 and 1), then applying the Pythagorean theorem and definitions of the trigonometric ratios. Purelyalgebraic derivations are longer.

3.2.3 Relationships among the inverse trigonometric functions

Complementary angles:

arccosx =π

2− arcsinx

arccotx =π

2− arctanx

arccscx =π

2− arcsecx

Negative arguments:

arcsin(−x) = − arcsinxarccos(−x) = π − arccosxarctan(−x) = − arctanxarccot(−x) = π − arccotxarcsec(−x) = π − arcsecxarccsc(−x) = − arccscx

Reciprocal arguments:

arccos(1/x) = arcsecxarcsin(1/x) = arccscx

arctan(1/x) = π

2− arctanx = arccotx , if x > 0

arctan(1/x) = −π

2− arctanx = arccotx− π , if x < 0

arccot(1/x) = π

2− arccotx = arctanx , if x > 0

arccot(1/x) = 3π

2− arccotx = π + arctanx , if x < 0

arcsec(1/x) = arccosxarccsc(1/x) = arcsinx

24 CHAPTER 3. INVERSE TRIGONOMETRIC FUNCTIONS

If you only have a fragment of a sine table:

arccosx = arcsin√1− x2 , if 0 ≤ x ≤ 1

arctanx = arcsin x√x2 + 1

Whenever the square root of a complex number is used here, we choose the root with the positive real part (or positiveimaginary part if the square was negative real).From the half-angle formula, tan θ

2 = sin θ1+cos θ , we get:

arcsinx = 2 arctan x

1 +√1− x2

arccosx = 2 arctan√1− x2

1 + x, if − 1 < x ≤ +1

arctanx = 2 arctan x

1 +√1 + x2

3.2.4 Arctangent addition formula

arctanu+ arctan v = arctan(

u+ v

1− uv

)(mod π) , uv ̸= 1 .

This is derived from the tangent addition formula

tan(α+ β) =tanα+ tanβ1− tanα tanβ ,

by letting

α = arctanu , β = arctan v .

3.3 In calculus

3.3.1 Derivatives of inverse trigonometric functionsMain article: Differentiation of trigonometric functions

The derivatives for complex values of z are as follows:

ddz arcsin z =

1√1− z2

; z ̸= −1,+1

ddz arccos z = − 1√

1− z2; z ̸= −1,+1

ddz arctan z =

1

1 + z2; z ̸= −i,+i

ddz arccot z = − 1

1 + z2; z ̸= −i,+i

ddz arcsec z =

1

z2√1− 1/z2

; z ̸= −1, 0,+1

ddz arccsc z = − 1

z2√1− 1/z2

; z ̸= −1, 0,+1

3.3. IN CALCULUS 25

Only for real values of x:

ddx arcsecx =

1

|x|√x2 − 1

; |x| > 1

ddx arccscx = − 1

|x|√x2 − 1

; |x| > 1

For a sample derivation: if θ = arcsinx , we get:

d arcsinxdx =

dθd sin θ =

dθcos θdθ =

1

cos θ =1√

1− sin2 θ=

1√1− x2

3.3.2 Expression as definite integrals

Integrating the derivative and fixing the value at one point gives an expression for the inverse trigonometric functionas a definite integral:

arcsinx =

∫ x

0

1√1− z2

dz , |x| ≤ 1

arccosx =

∫ 1

x

1√1− z2

dz , |x| ≤ 1

arctanx =

∫ x

0

1

z2 + 1dz ,

arccotx =

∫ ∞

x

1

z2 + 1dz ,

arcsecx =

∫ x

1

1

z√z2 − 1

dz = π +

∫ −1

x

1

z√z2 − 1

dz , x ≥ 1

arccscx =

∫ ∞

x

1

z√z2 − 1

dz =

∫ x

−∞

1

z√z2 − 1

dz , x ≥ 1

When x equals 1, the integrals with limited domains are improper integrals, but still well-defined.

3.3.3 Infinite series

Like the sine and cosine functions, the inverse trigonometric functions can be calculated using power series, as follows.For arcsine, the series can be derived by expanding its derivative, 1√

1−z2, as a binomial series, and integrating term

by term (using the integral definition as above). The series for arctangent can similarly be derived by expanding itsderivative 1

1+z2 in a geometric series and applying the integral definition above (see Leibniz series).

arcsin z = z +

(1

2

)z3

3+

(1 · 32 · 4

)z5

5+

(1 · 3 · 52 · 4 · 6

)z7

7+ · · · =

∞∑n=0

(2nn

)z2n+1

4n(2n+ 1); |z| ≤ 1

arccos z =π

2− arcsin z =

π

2−(z +

(1

2

)z3

3+

(1 · 32 · 4

)z5

5+ · · ·

)=

π

2−

∞∑n=0

(2nn

)z2n+1

4n(2n+ 1); |z| ≤ 1

arctan z = z − z3

3+

z5

5− z7

7+ · · · =

∞∑n=0

(−1)nz2n+1

2n+ 1; |z| ≤ 1 z ̸= i,−i

arccot z =π

2−arctan z =

π

2−(z − z3

3+

z5

5− z7

7+ · · ·

)=

π

2−

∞∑n=0

(−1)nz2n+1

2n+ 1; |z| ≤ 1 z ̸= i,−i

26 CHAPTER 3. INVERSE TRIGONOMETRIC FUNCTIONS

arcsec z = arccos(1/z) = π

2−(z−1 +

(1

2

)z−3

3+

(1 · 32 · 4

)z−5

5+ · · ·

)=

π

2−

∞∑n=0

(2nn

)z−(2n+1)

4n(2n+ 1); |z| ≥ 1

arccsc z = arcsin(1/z) = z−1 +

(1

2

)z−3

3+

(1 · 32 · 4

)z−5

5+ · · · =

∞∑n=0

(2nn

)z−(2n+1)

4n(2n+ 1); |z| ≥ 1

Leonhard Euler found a more efficient series for the arctangent, which is:

arctan z =z

1 + z2

∞∑n=0

n∏k=1

2kz2

(2k + 1)(1 + z2).

(Notice that the term in the sum for n = 0 is the empty product which is 1.)Alternatively, this can be expressed:

arctan z =∞∑

n=0

22n(n!)2

(2n+ 1)!

z2n+1

(1 + z2)n+1

Variant: Continued fractions for arctangent

Two alternatives to the power series for arctangent are these generalized continued fractions:

arctan z =z

1 +(1z)2

3− 1z2 +(3z)2

5− 3z2 +(5z)2

7− 5z2 +(7z)2

9− 7z2 +. . .

=z

1 +(1z)2

3 +(2z)2

5 +(3z)2

7 +(4z)2

9 +. . .

The second of these is valid in the cut complex plane. There are two cuts, from −i to the point at infinity, going downthe imaginary axis, and from i to the point at infinity, going up the same axis. It works best for real numbers runningfrom −1 to 1. The partial denominators are the odd natural numbers, and the partial numerators (after the first) arejust (nz)2, with each perfect square appearing once. The first was developed by Leonhard Euler; the second by CarlFriedrich Gauss utilizing the Gaussian hypergeometric series.

3.3.4 Indefinite integrals of inverse trigonometric functions

For real and complex values of z:

∫arcsin z dz = z arcsin z +

√1− z2 + C∫

arccos z dz = z arccos z −√

1− z2 + C∫arctan z dz = z arctan z − 1

2ln(1 + z2

)+ C∫

arccot z dz = z arccot z + 1

2ln(1 + z2

)+ C∫

arcsec z dz = z arcsec z − ln[z

(1 +

√z2 − 1

z2

)]+ C

∫arccsc z dz = z arccsc z + ln

[z

(1 +

√z2 − 1

z2

)]+ C

3.3. IN CALCULUS 27

For real x ≥ 1:

∫arcsecx dx = x arcsecx− ln

(x+

√x2 − 1

)+ C∫

arccscx dx = x arccscx+ ln(x+

√x2 − 1

)+ C

For all real x not between −1 and 1:

∫arcsecx dx = x arcsecx− sgn(x) ln

∣∣∣x+√

x2 − 1∣∣∣+ C∫

arccscx dx = x arccscx+ sgn(x) ln∣∣∣x+

√x2 − 1

∣∣∣+ C

The absolute value is necessary to compensate for both negative and positive values of the arcsecant and arccosecantfunctions. The signum function is also necessary due to the absolute values in the derivatives of the two functions,which create two different solutions for positive and negative values of x. These can be further simplified using thelogarithmic definitions of the inverse hyperbolic functions:

∫arcsecx dx = x arcsecx− arcosh |x|+ C∫arccscx dx = x arccscx+ arcosh |x|+ C

The absolute value in the argument of the arcosh function creates a negative half of its graph, making it identical tothe signum logarithmic function shown above.All of these antiderivatives can be derived using integration by parts and the simple derivative forms shown above.

Example

Using∫u dv = uv −

∫v du , set

u = arcsinx dv = dx

du =dx√1− x2

v = x

Then

∫arcsin(x) dx = x arcsinx−

∫x√

1− x2dx

Substitute

w = 1− x2 .

Then

dw = −2x dx

and

28 CHAPTER 3. INVERSE TRIGONOMETRIC FUNCTIONS

∫x√

1− x2dx = −1

2

∫ dw√w

= −√w

Back-substitute for x to yield

∫arcsin(x) dx = x arcsinx+

√1− x2 + C

3.4 Extension to complex plane

Since the inverse trigonometric functions are analytic functions, they can be extended from the real line to the complexplane. This results in functions with multiple sheets and branch points. One possible way of defining the extensionsis:

arctan z =

∫ z

0

dx1 + x2

z ̸= −i,+i

where the part of the imaginary axis which does not lie strictly between −i and +i is the cut between the principalsheet and other sheets;

arcsin z = arctan z√1− z2

z ̸= −1,+1

where (the square-root function has its cut along the negative real axis and) the part of the real axis which does notlie strictly between −1 and +1 is the cut between the principal sheet of arcsin and other sheets;

arccos z =π

2− arcsin z z ̸= −1,+1

which has the same cut as arcsin;

arccot z =π

2− arctan z z ̸= −i,+i

which has the same cut as arctan;

arcsec z = arccos 1z

z ̸= −1, 0,+1

where the part of the real axis between −1 and +1 inclusive is the cut between the principal sheet of arcsec and othersheets;

arccsc z = arcsin 1

zz ̸= −1, 0,+1

which has the same cut as arcsec.

3.4.1 Logarithmic forms

These functions may also be expressed using complex logarithms. This extends in a natural fashion their domain tothe complex plane.

3.4. EXTENSION TO COMPLEX PLANE 29

arcsin z = −i ln(iz +

√1− z2

)= arccsc 1

z

arccos z = −i ln(z +

√z2 − 1

)=

π

2+ i ln

(iz +

√1− z2

)=

π

2− arcsin z = arcsec 1

z

arctan z = 12 i [ln (1− iz)− ln (1 + iz)] = arccot 1

z

arccot z = 12 i

[ln(1− i

z

)− ln

(1 +

i

z

)]= arctan 1

z

arcsec z = −i ln(√

1

z2− 1 +

1

z

)= i ln

(√1− 1

z2+

i

z

)+

π

2=

π

2− arccsc z = arccos 1

z

arccsc z = −i ln(√

1− 1

z2+

i

z

)= arcsin 1

z

Elementary proofs of these relations proceed via expansion to exponential forms of the trigonometric functions.

Example proof

sin(ϕ) = z

ϕ = arcsin zUsing the exponential definition of sine, one obtains

z =eiϕ − e−iϕ

2i

Let

ξ = eiϕ

Solving for ϕ

z =ξ − 1/ξ

2i

2iz = ξ − 1/ξ

ξ − 2iz − 1/ξ = 0

ξ2 − 2iξz − 1 = 0

ξ = iz ±√1− z2 = eiϕ

iϕ = ln(iz ±

√1− z2

)ϕ = −i ln

(iz ±

√1− z2

)(the positive branch is chosen)

ϕ = arcsin z = −i ln(iz +

√1− z2

)

30 CHAPTER 3. INVERSE TRIGONOMETRIC FUNCTIONS

3.5 Applications

3.5.1 General solutions

Each of the trigonometric functions is periodic in the real part of its argument, running through all its values twice ineach interval of 2π. Sine and cosecant begin their period at 2πk − π/2 (where k is an integer), finish it at 2πk + π/2,and then reverse themselves over 2πk + π/2 to 2πk + 3π/2. Cosine and secant begin their period at 2πk, finish it at2πk + π, and then reverse themselves over 2πk + π to 2πk + 2π. Tangent begins its period at 2πk − π/2, finishes it at2πk + π/2, and then repeats it (forward) over 2πk + π/2 to 2πk + 3π/2. Cotangent begins its period at 2πk, finishesit at 2πk + π, and then repeats it (forward) over 2πk + π to 2πk + 2π.This periodicity is reflected in the general inverses where k is some integer:

sin(y) = x ⇔ y = arcsin(x) + 2πk or y = π − arcsin(x) + 2πk

sin(y) = x ⇔ y = (−1)k arcsin(x) + πk

cos(y) = x ⇔ y = arccos(x) + 2πk or y = 2π − arccos(x) + 2πk

cos(y) = x ⇔ y = ± arccos(x) + 2πk

tan(y) = x ⇔ y = arctan(x) + πk

cot(y) = x ⇔ y = arccot(x) + πk

sec(y) = x ⇔ y = arcsec(x) + 2πk or y = 2π − arcsec(x) + 2πk

csc(y) = x ⇔ y = arccsc(x) + 2πk or y = π − arccsc(x) + 2πk

Application: finding the angle of a right triangle

Inverse trigonometric functions are useful when trying to determine the remaining two angles of a right triangle whenthe lengths of the sides of the triangle are known. Recalling the right-triangle definitions of sine, for example, itfollows that

θ = arcsin( oppositehypotenuse

).

Often, the hypotenuse is unknown and would need to be calculated before using arcsine or arccosine using thePythagorean Theorem: a2 + b2 = h2 where h is the length of the hypotenuse. Arctangent comes in handy inthis situation, as the length of the hypotenuse is not needed.

θ = arctan(oppositeadjacent

).

For example, suppose a roof drops 8 feet as it runs out 20 feet. The roof makes an angle θ with the horizontal, whereθ may be computed as follows:

θ = arctan(oppositeadjacent

)= arctan

( riserun

)= arctan

(8

20

)≈ 21.8◦ .

3.5.2 In computer science and engineering

3.6. SEE ALSO 31

Two-argument variant of arctangent

Main article: atan2

The two-argument atan2 function computes the arctangent of y / x given y and x, but with a range of (−π, π]. Inother words, atan2(y, x) is the angle between the positive x-axis of a plane and the point (x, y) on it, with positivesign for counter-clockwise angles (upper half-plane, y > 0), and negative sign for clockwise angles (lower half-plane,y < 0). It was first introduced in many computer programming languages, but it is now also common in other fieldsof science and engineering.In terms of the standard arctan function, that is with range of (−π/2, π/2), it can be expressed as follows:

atan2(y, x) =

arctan( yx ) x > 0

arctan( yx ) + π y ≥ 0 , x < 0

arctan( yx )− π y < 0 , x < 0π2 y > 0 , x = 0

−π2 y < 0 , x = 0

undefined y = 0 , x = 0

It also equals the principal value of the argument of the complex number x + iy.This function may also be defined using the tangent half-angle formulae as follows:

atan2(y, x) = 2 arctan y√x2 + y2 + x

provided that either x > 0 or y ≠ 0. However this fails if given x ≤ 0 and y = 0 so the expression is unsuitable forcomputational use.The above argument order (y, x) seems to be the most common, and in particular is used in ISO standards such asthe C programming language, but a few authors may use the opposite convention (x, y) so some caution is warranted.These variations are detailed at atan2.

Arctangent function with location parameter

In many applications the solution y of the equation x = tan y is to come as close as possible to a given value−∞ < η < ∞ . The adequate solution is produced by the parameter modified arctangent function

y = arctanη x := arctanx+ π · rni η − arctanxπ

.

The function rni rounds to the nearest integer.

Practical considerations

For angles near 0 and π, arccosine is ill-conditioned and will thus calculate the angle with reduced accuracy in acomputer implementation (due to the limited number of digits). Similarly, arcsine is inaccurate for angles near −π/2and π/2. To achieve full accuracy for all angles, arctangent or atan2 should be used for the implementation.

3.6 See also

• Argument (complex analysis)

• Complex logarithm

32 CHAPTER 3. INVERSE TRIGONOMETRIC FUNCTIONS

• Gauss’s continued fraction

• Inverse hyperbolic function

• List of integrals of inverse trigonometric functions

• List of trigonometric identities

• Square root

• Tangent half-angle formula

• Trigonometric function

3.7 References[1] For example Dörrie, Heinrich (1965). Triumph der Mathematik. Trans. David Antin. Dover. p. 69. ISBN 0-486-61348-8.

[2] Prof. Sanaullah Bhatti; Ch. Nawab-ud-Din; Ch. Bashir Ahmed; Dr. S. M. Yousuf; Dr. Allah Bukhsh Taheem (1999).“Differentiation of Tigonometric, Logarithmic and Exponential Functions”. In Prof. Mohammad Maqbool Ellahi, Dr.Karamat Hussain Dar, Faheem Hussain. Calculus and Analytic Geometry (in Pakistani English) (First ed.). Lahore: PunjabTextbook Board. p. 140.

[3] Cajori, Florian (1919). A History of Mathematics (2nd ed.). The Macmillan Company, New York. p. 272., at GoogleBooks

[4] Herschel, John F. W. (1813). “On a remarkable Application of Cotes’s Theorem”. Philosophical Transactions (RoyalSociety, London) 103 (1): 10., at Google Books

[5] “Inverse trigonometric functions” in The Americana: a universal reference library, Vol.21, Ed. Frederick Converse Beach,George Edwin Rines, (1912).

3.8 External links• Weisstein, Eric W., “Inverse Trigonometric Functions”, MathWorld.

• Weisstein, Eric W., “Inverse Tangent”, MathWorld.

3.8. EXTERNAL LINKS 33

The usual principal values of the arcsin(x) (red) and arccos(x) (blue) functions graphed on the cartesian plane.

34 CHAPTER 3. INVERSE TRIGONOMETRIC FUNCTIONS

The usual principal values of the arctan(x) and arccot(x) functions graphed on the cartesian plane.

Principal values of the arcsec(x) and arccsc(x) functions graphed on the cartesian plane.

3.8. EXTERNAL LINKS 35

A C

B

b

ah

(adjacent)

(opposite)(hypotenuse)

A right triangle.

Chapter 4

Multiplicative inverse

-5

-4

-3

-2

-1

0

1

2

3

4

5

-5 -4 -3 -2 -1 0 1 2 3 4 5

1 x

The reciprocal function: y = 1/ x. For every x except 0, y represents its multiplicative inverse. The graph forms a rectangularhyperbola.

In mathematics, a multiplicative inverse or reciprocal for a number x, denoted by 1/x or x−1, is a number whichwhen multiplied by x yields the multiplicative identity, 1. The multiplicative inverse of a fraction a/b is b/a. For themultiplicative inverse of a real number, divide 1 by the number. For example, the reciprocal of 5 is one fifth (1/5 or0.2), and the reciprocal of 0.25 is 1 divided by 0.25, or 4. The reciprocal function, the function f(x) that maps x to1/x, is one of the simplest examples of a function which is its own inverse (an involution).The term reciprocal was in common use at least as far back as the third edition of Encyclopædia Britannica (1797)to describe two numbers whose product is 1; geometrical quantities in inverse proportion are described as reciprocallin a 1570 translation of Euclid's Elements.[1]

In the phrasemultiplicative inverse, the qualifiermultiplicative is often omitted and then tacitly understood (in contrastto the additive inverse). Multiplicative inverses can be defined over many mathematical domains as well as numbers.

36

4.1. EXAMPLES AND COUNTEREXAMPLES 37

In these cases it can happen that ab ≠ ba; then “inverse” typically implies that an element is both a left and rightinverse.The notation f −1 is sometimes also used for the inverse function of the function f, which is not in general equal tothe multiplicative inverse. For example, the multiplicative inverse 1/(sin x) = (sin x)−1 is different from the inversesin of x, denoted sin−1 x or arcsin x. Only for linear maps are they strongly related (see below). The terminologydifference reciprocal versus inverse is not sufficient to make this distinction, since many authors prefer the oppositenaming convention, probably for historical reasons (for example in French, the inverse function is preferably calledapplication réciproque).

4.1 Examples and counterexamples

In the real numbers, zero does not have a reciprocal because no real number multiplied by 0 produces 1 (the productof any number with zero is zero). With the exception of zero, reciprocals of every real number are real, reciprocalsof every rational number are rational, and reciprocals of every complex number are complex. The property that everyelement other than zero has a multiplicative inverse is part of the definition of a field, of which these are all examples.On the other hand, no integer other than 1 and −1 has an integer reciprocal, and so the integers are not a field.In modular arithmetic, the modular multiplicative inverse of a is also defined: it is the number x such that ax ≡ 1(mod n). This multiplicative inverse exists if and only if a and n are coprime. For example, the inverse of 3 modulo11 is 4 because 4 · 3 ≡ 1 (mod 11). The extended Euclidean algorithm may be used to compute it.The sedenions are an algebra in which every nonzero element has a multiplicative inverse, but which nonetheless hasdivisors of zero, i.e. nonzero elements x, y such that xy = 0.A square matrix has an inverse if and only if its determinant has an inverse in the coefficient ring. The linear mapthat has the matrix A−1 with respect to some base is then the reciprocal function of the map having A as matrix inthe same base. Thus, the two distinct notions of the inverse of a function are strongly related in this case, while theymust be carefully distinguished in the general case (as noted above).The trigonometric functions are related by the reciprocal identity: the cotangent is the reciprocal of the tangent; thesecant is the reciprocal of the cosine; the cosecant is the reciprocal of the sine.A ring in which every nonzero element has a multiplicative inverse is a division ring; likewise an algebra in which thisholds is a division algebra.

4.2 Complex numbers

As mentioned above, the reciprocal of every nonzero complex number z = a + bi is complex. It can be found bymultiplying both top and bottom of 1/z by its complex conjugate z̄ = a− bi and using the property that zz̄ = ∥z∥2, the absolute value of z squared, which is the real number a2 + b2:

1

z=

z̄

zz̄=

z̄

∥z∥2=

a− bi

a2 + b2=

a

a2 + b2− b

a2 + b2i.

In particular, if ||z||=1 (z has unit magnitude), then 1/z = z̄ . Consequently, the imaginary units, ±i, have additiveinverse equal to multiplicative inverse, and are the only complex numbers with this property. For example, additiveand multiplicative inverses of i are −(i) = −i and 1/i = −i, respectively.For a complex number in polar form z = r(cos φ + i sin φ), the reciprocal simply takes the reciprocal of the magnitudeand the negative of the angle:

1

z=

1

r(cos(−φ) + i sin(−φ)) .

38 CHAPTER 4. MULTIPLICATIVE INVERSE

0 1 2 3 4 5 6 7 8

0.3 0.6 0.9 1.2 1.5 1.8 2.1

Geometric intuition for the integral of 1/ x. The three integrals from 1 to 2, from 2 to 4, and from 4 to 8 are all equal. Each regionis the previous region scaled vertically down by 50%, then horizontally by 200%. Extending this, the integral from 1 to 2k is k timesthe integral from 1 to 2, just as ln 2k = k ln 2.

4.3 Calculus

In real calculus, the derivative of 1/x = x−1 is given by the power rule with the power −1:

d

dxx−1 = (−1)x(−1)−1 = −x−2 = − 1

x2.

The power rule for integrals (Cavalieri’s quadrature formula) cannot be used to compute the integral of 1/x, becausedoing so would result in division by 0:

∫1

xdx =

x0

0+ C

Instead the integral is given by:

∫ a

1

1

xdx = ln a,

4.4. ALGORITHMS 39

∫1

xdx = lnx+ C.

where ln is the natural logarithm. To show this, note that ddxe

x = ex , so if y = ex and x = ln y , we have:[2]

dy

dx= y ⇒ dy

y= dx ⇒

∫1

ydy =

∫1 dx ⇒

∫1

ydy = x+ C = ln y + C.

4.4 Algorithms

The reciprocal may be computed by hand with the use of long division.Computing the reciprocal is important in many division algorithms, since the quotient a/b can be computed by firstcomputing 1/b and then multiplying it by a. Noting that f(x) = 1/x− b has a zero at x = 1/b, Newton’s method canfind that zero, starting with a guess x0 and iterating using the rule:

xn+1 = xn − f(xn)

f ′(xn)= xn − 1/xn − b

−1/x2n

= 2xn − bx2n = xn(2− bxn).

This continues until the desired precision is reached. For example, suppose we wish to compute 1/17 ≈ 0.0588 with3 digits of precision. Taking x0 = 0.1, the following sequence is produced:

x1 = 0.1(2 - 17 × 0.1) = 0.03x2 = 0.03(2 - 17 × 0.03) = 0.0447x3 = 0.0447(2 - 17 × 0.0447) ≈ 0.0554x4 = 0.0554(2 - 17 × 0.0554) ≈ 0.0586x5 = 0.0586(2 - 17 × 0.0586) ≈ 0.0588

A typical initial guess can be found by rounding b to a nearby power of 2, then using bit shifts to compute its reciprocal.In constructive mathematics, for a real number x to have a reciprocal, it is not sufficient that x ≠ 0. There must insteadbe given a rational number r such that 0 < r < |x|. In terms of the approximation algorithm described above, this isneeded to prove that the change in y will eventually become arbitrarily small.This iteration can also be generalised to a wider sort of inverses, e.g. matrix inverses.

4.5 Reciprocals of irrational numbers

Every number excluding zero has a reciprocal, and reciprocals of certain irrational numbers can have importantspecial properties. Examples include the reciprocal of e (≈ 0.367879)and the golden ratio’s reciprocal (≈ 0.618034).The first reciprocal is special because no other positive number can produce a lower number when put to the powerof itself; f(1/e) is the global minimum of f(x) = xx . The second number is the only positive number that is equalto its reciprocal plus one: ϕ = 1/ϕ+1 . Its additive inverse is the only negative number that is equal to its reciprocalminus one: −ϕ = −1/ϕ− 1 .The function f(n) = n+

√(n2 + 1), n ∈ N,n > 0 gives an infinite number of irrational numbers that differ with

their reciprocal by an integer. For example, f(2) is the irrational 2 +√5 . Its reciprocal 1/(2 +

√5) is −2 +

√5 ,

exactly 4 less. Such irrational numbers share a curious property: they have the same fractional part as their reciprocal.

4.6 Further remarks

If the multiplication is associative, an element x with a multiplicative inverse cannot be a zero divisor (meaning forsome y, xy = 0 with neither x nor y equal to zero). To see this, it is sufficient to multiply the equation xy = 0 by the

40 CHAPTER 4. MULTIPLICATIVE INVERSE

0.2 0.4 0.6 0.8 1.0 1.2 1.4 0.0

0.5

1.0

1.5

Graph of f(x) = xx showing the minimum at (1/e, e−1/e).

inverse of x (on the left), and then simplify using associativity. In the absence of associativity, the sedenions providea counterexample.The converse does not hold: an element which is not a zero divisor is not guaranteed to have a multiplicative inverse.Within Z, all integers except −1, 0, 1 provide examples; they are not zero divisors nor do they have inverses in Z. Ifthe ring or algebra is finite, however, then all elements a which are not zero divisors do have a (left and right) inverse.For, first observe that the map ƒ(x) = ax must be injective: ƒ(x) = ƒ(y) implies x = y:

ax = ay ⇒ ax− ay = 0

⇒ a(x− y) = 0

⇒ x− y = 0

⇒ x = y.

Distinct elements map to distinct elements, so the image consists of the same finite number of elements, and the mapis necessarily surjective. Specifically, ƒ (namely multiplication by a) must map some element x to 1, ax = 1, so that xis an inverse for a.

4.7 Applications

The expansion of the reciprocal 1/q in any base can also act [3] as a source of pseudo-random numbers, if q is a“suitable” safe prime, a prime of the form 2p + 1 where p is also a prime. A sequence of pseudo-random numbersof length q − 1 will be produced by the expansion.

4.8 See also• Division (mathematics)

4.9. NOTES 41

• Fraction (mathematics)

• Group (mathematics)

• Ring (mathematics)

• Division algebra

• Exponential decay

• Unit fractions – reciprocals of integers

• Hyperbola

• Repeating decimal

• List of sums of reciprocals

4.9 Notes[1] " In equall Parallelipipedons the bases are reciprokall to their altitudes”. OED “Reciprocal” §3a. Sir Henry Billingsley

translation of Elements XI, 34.

[2] Anthony, Dr. “Proof that INT(1/x)dx = lnx”. Ask Dr. Math. Drexel University. Retrieved 22 March 2013.

[3] Mitchell, Douglas W., “A nonlinear random number generator with known, long cycle length,” Cryptologia 17, January1993, 55-62.

4.10 References• Maximally Periodic Reciprocals, Matthews R.A.J. Bulletin of the Institute of Mathematics and its Applicationsvol 28 pp 147–148 1992

42 CHAPTER 4. MULTIPLICATIVE INVERSE

4.11 Text and image sources, contributors, and licenses

4.11.1 Text

• Function composition Source: https://en.wikipedia.org/wiki/Function_composition?oldid=680054219 Contributors: Zundark, Tarquin,Patrick, Michael Hardy, Wshun, Kku, Dcljr, TakuyaMurata, Glenn, Andres, Charles Matthews, Timwi, Greenrd, Phys, Phil Boswell,Robbot, Rasmus Faber, Tobias Bergemann, Giftlite, Lethe, MSGJ, Dratman, Jason Quinn, Macrakis, Rheun, Karl Dickman, Paul August,Danakil, EmilJ, HasharBot~enwiki, Oleg Alexandrov, Woohookitty, Linas, Georgia guy, Zenkat, MattGiuca, Mpatel, MFH, GregorB,Qwertyus, SixWingedSeraph, Rjwilmsi, Slac, FlaBot, VKokielov, Vonkje, YurikBot, Grubber, NawlinWiki, Googl, Netrapt, Maksim-e~enwiki, Adammajewski, Incnis Mrsi, Melchoir, XudongGuan~enwiki, BiT, Nbarth, Javalenok, J•A•K, SundarBot, Ecsnp, Jon Awbrey,Lambiam, Dmh~enwiki, DA3N, Dfass, EdC~enwiki, Cherry Cotton, CBM, Strangelv, Juansempere, Escarbot, QuiteUnusual, Kuteni,JAnDbot, Gcm, Fuzzybyte, Magioladitis, David Eppstein, Policron, Cuzkatzimhut, VolkovBot, Pleasantville, JohnBlackburne, LokiClock,TXiKiBoT, Anonymous Dissident, Jonnyappleseed24, TrippingTroubadour, AlleborgoBot, EmxBot, Pit-trout, Classicalecon, ClueBot,Marino-slo, Plastikspork, SoxBot III, XLinkBot, Addbot, Lightbot, PV=nRT, Luckas-bot, Pcap, KamikazeBot, Zubachi, LilHelpa, Xqbot,RibotBOT, Einkil, Constructive editor, Pinethicket, RedBot, Fallenness, John of Reading, ZéroBot, Quondum, Aughost, EdoBot, ClueBotNG,Wcherowi, Aurelian Radoaca, Gauravjuvekar, Shashank rathore, Brad7777, ChrisGualtieri, JamesHaigh, Mathdiskteacher, Makecat-bot, Imareaver, Jochen Burghardt, Limit-theorem,Monkbot, Ashleyelizabethmath4626, Lkmhokie8, JMPEAX, Dchsnq andAnonymous:67

• Inverse function Source: https://en.wikipedia.org/wiki/Inverse_function?oldid=682224507 Contributors: AxelBoldt, Tarquin, Bdesham,Michael Hardy, Looxix~enwiki, Glenn, Poor Yorick, Charles Matthews, Dcoetzee, Dysprosia, Jitse Niesen, Greenrd, Saltine, Kwantus,Mina86, Gromlakh, Robbot, Fredrik, Scarfboy, Bkell, Tobias Bergemann, Tosha, Giftlite, Qartis, John Palkovic, Icairns, PhotoBox,Discospinster, Guanabot, ObsessiveMathsFreak, ReiVaX, Paul August, Kenb215, El C, Passw0rd, EvenT, Infobacker, Oleg Alexandrov,Woohookitty, MFH, Ryan Reich, Graham87, Qwertyus, Rjwilmsi, JVz, Salix alba, VKokielov, ChongDae, CiaPan, Chobot, Krish-navedala, YurikBot, Wavelength, KSmrq, Rick Norwood, Bota47, Kompik, Gesslein, Banus, SmackBot, Incnis Mrsi, Eskimbot, Xaos-flux, Nbarth, SundarBot, Wen D House, Jon Awbrey, Lambiam, Jim.belk, Dr Greg, Hvn0413, Dreftymac, Happy-melon, JRSpriggs,Arabic Pilot, Runningonbrains, Mct mht, Gregbard, Sam Staton, Blindman shady, Thijs!bot, Kilva, Lt. CiberShark~enwiki, Paquitotrek,EdJohnston, Escarbot, TK-925, Béka, Gcm, Unifey~enwiki, Burga, VoABot II, Tibordp, Planemo, Jwuthe2, Haseldon, Policron, Lo-kiClock, PMajer, Anonymous Dissident, Clark Kimberling, Broadbot, Synthebot, AlleborgoBot, EverGreg, EmxBot, Ken Kuniyuki,Quietbritishjim, SieBot, Ivan Štambuk, RJaguar3, Paolo.dL, Harry~enwiki, Randomblue, Wahrmund, ClueBot, Rustic, Marino-slo,The Thing That Should Not Be, Ldimicco, Mild Bill Hiccup, Excirial, Zlewis101, Addbot, Download, Ehrenkater, Lightbot, Jarble,Legobot, Luckas-bot, Yobot, Fraggle81, Legobot II, KamikazeBot, AnomieBOT, JackieBot, Farhil, Hombre1729, Flewis, E2eamon,Diego Queiroz, DannyAsher, Xqbot, Sionus, NOrbeck, SassoBot, ViolaPlayer, FrescoBot, OgreBot, 00Ragora00, I dream of horses,Belovedeagle, Suffusion of Yellow, Hrvatistan, DASHBot, EmausBot, RA0808, Wikipelli, RaptureBot, Anti-min, Chewings72, ClueBotNG, Wcherowi, Matthiaspaul, Escapepea, Helpful Pixie Bot, Leonxlin, MadamIamadam, Frze, Matha288, JayEB, Brad7777, Mogism,PC-XT, Sogenius, DavidLeighEllis, Werddemer, Danielsevero, Wikiyroc, Kkim10, Fmmmlee, Philologick and Anonymous: 163

• Inverse trigonometric functions Source: https://en.wikipedia.org/wiki/Inverse_trigonometric_functions?oldid=684105717 Contribu-tors: XJaM, Patrick, Michael Hardy, Stevenj, Dysprosia, Fibonacci, Robbot, Tobias Bergemann, Giftlite, Anville, SURIV, Daniel,levine,Pmanderson, Abdull, Discospinster, Osrevad, Zenohockey, Army1987, Alansohn, Anthony Appleyard, Wtmitchell, StradivariusTV, Ar-mando, Gerbrant, Emallove, R.e.b., Kri, Glenn L, Salvatore Ingala, Chobot, Visor, DVdm, Algebraist, YurikBot, Wavelength, Sceptre,Hede2000, KSmrq, Grafen, Int 80h, NorsemanII, Bamse, RDBury, Maksim-e~enwiki, Thelukeeffect, Eskimbot, Mhss, Mirokado, JC-Santos, PrimeHunter, Deathanatos, V1adis1av, Saippuakauppias, Lambiam, Eridani, Ian Vaughan, ChaoticLlama, CapitalR, JRSpriggs,Conrad.Irwin, HenningThielemann, Fommil, Rian.sanderson, Palmtree3000, Zalgo, Thijs!bot, Spikedmilk, Nonagonal Spider, EdJohn-ston, Luna Santin, Hannes Eder, JAnDbot, Ricardo sandoval, Jetstreamer, JNW, Albmont, Gamkiller, JoergenB, Ac44ck, Gwern,Isamil, Mythealias, Pomte, Knorlin, TungstenWolfram, Hennessey, Patrick, Bobianite, BentonMiller, Sigmundur, DavidCBryant, AlanU. Kennington, VolkovBot, Indubitably, LokiClock, Justtysen, VasilievVV, Riku92mr, Anonymous Dissident, Corvus coronoides, Dmcq,Vertciel, Logan, CagedKiller360, Aly89, AlanUS, ClueBot, JoeHillen, Bender2k14, Cenarium, Leandropls, Kiensvay, Nikhilkrgvr,Aaron north, Dthomsen8, DaL33T, Addbot, Fgnievinski, Iceblock, Zarcadia, Sleepaholic, Jasper Deng, Zorrobot, Luckas-bot, Yobot,Tohd8BohaithuGh1, Ptbotgourou, TaBOT-zerem, AnomieBOT, JackieBot, Nickweedon, Geek1337~enwiki, Diego Queiroz, Txebixev,St.nerol, Hdullin, GrouchoBot, Uniwersalista, SassoBot, Prari, Nixphoeni, D'ohBot, Kusluj, Emjayeff, Number Googol, Adammer-linsmith, TjBot, Jowa fan, EmausBot, ModWilson, Velowiki, X-4-V-I, Wham Bam Rock II, ZéroBot, Michael.YX.Wu, Isaac Euler,Tolly4bolly, Jay-Sebastos, Colin.campbell.27, Maschen, ChuispastonBot, Rmashhadi, ClueBot NG, Hdreuter, Helpful Pixie Bot, KLBot2,Vagobot, Crh23, YatharthROCK, StevinSimon, Tfr000, Modalanalytiker, Pratyya Ghosh, Ahmed Magdy Hosny, Brirush, Yardimsever,Wamiq, Jerming, Blackbombchu, Pqnlrn, DTL LAPOS, De Riban5, Monkbot, Arsenal CR7, Cdserio99 and Anonymous: 171

• Multiplicative inverse Source: https://en.wikipedia.org/wiki/Multiplicative_inverse?oldid=682637979Contributors: TobyBartels, Patrick,Michael Hardy, Ixfd64, Eric119, Glenn, Timwi, Dcoetzee, Frazzydee, Robbot, Giftlite, Fropuff, No Guru, Pmanderson, TrevorMacInnis,Chrisjwmartin, Discospinster, Rich Farmbrough, ArnoldReinhold, Notthepainter, EmilJ, Bobo192, Wood Thrush, Haham hanuka, Jum-buck, Alansohn, Tobych, Jheald, Natalya, Georgia guy, Ruud Koot, WadeSimMiser, MFH, Marudubshinki, Dysepsion, Jshadias, JoshParris, Mathbot, Chobot, Jersey Devil, Algebraist, Pandelon, RussBot, Michael Slone, KSmrq, Goffrie, Zwobot, Bota47, Wknight94,Square87~enwiki, Arthur Rubin, JLaTondre, Gesslein, Leon2323, Ghazer~enwiki, Bo Jacoby, KocjoBot~enwiki, GraemeMcRae, DanHoey, Octahedron80, HoodedMan, UU, Wen D House, Dreadstar, Tbjw, Rigadoun, Mgiganteus1, Jim.belk, Slakr, Mets501, AvantGuard, Levineps, Madmath789, CRGreathouse, CBM, OMGsplosion, Doctormatt, He Who Is, Omicronpersei8, Thijs!bot, Epbr123,FreeKresge, AntiVandalBot, Mhaitham.shammaa, Salgueiro~enwiki, TuvicBot, JAnDbot, Greensburger, Lawilkin, VoABot II, ToomaiGlittershine, MyNameIsNeo, David Eppstein, Rettetast, Anaxial, Leyo, Zorakoid, Uncle Dick, Mike.lifeguard, WarthogDemon, OohBun-nies!, Montchav, R00723r0, Philip Trueman, TXiKiBoT, Nxavar, Gauge00, Kmhkmh, Dmcq, Logan, Ben Boldt, Dogah, Caltas, Jackpots,Smoby10, Oxymoron83, JackSchmidt, ClueBot, Justin W Smith, Cb4astros, BarretB, Marc van Leeuwen, Rror, Coolbeans39, StephenPoppitt, Addbot, Ramu50, Friginator, Ronhjones, Glane23, Numbo3-bot, Tide rolls, Narnaja, Snaily, Luckas-bot, Yobot, PMLawrence,Piano non troppo, Kingpin13, Moipaulochon, DirlBot, Xqbot, RibotBOT, Mothernessfather, Robo37, Hoo man, RedBot, Duoduoduo,Copistopplayer, WikitanvirBot, ZéroBot, Quondum, Vanished user fijtji34toksdcknqrjn54yoimascj, NTox, Lyleq, ClueBot NG, HMSSo-lent, Calabe1992, AvocatoBot, Hillcrest98, Yomama719, Everymorning, Lordofbartonpark, Ginsuloft, Loraof and Anonymous: 167

4.11. TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES 43

4.11.2 Images• File:Absolute_value_composition.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/ba/Absolute_value_composition.

svg License: CC0 Contributors: Own work Original artist: Incnis Mrsi• File:Academ_Example_of_similarity_with_ratio_square_root_of_2.svg Source: https://upload.wikimedia.org/wikipedia/commons/

3/33/Academ_Example_of_similarity_with_ratio_square_root_of_2.svg License: CCBY-SA 3.0Contributors: OwnworkOriginal artist:Yves Baelde

• File:Ambox_important.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Ambox_important.svg License: Public do-main Contributors: Own work, based off of Image:Ambox scales.svg Original artist: Dsmurat (talk · contribs)

• File:Arcsecant_Arccosecant.svg Source: https://upload.wikimedia.org/wikipedia/commons/5/56/Arcsecant_Arccosecant.svg License:CC BY-SA 3.0 Contributors: Own work Original artist: Geek3

• File:Arcsine_Arccosine.svg Source: https://upload.wikimedia.org/wikipedia/commons/b/b4/Arcsine_Arccosine.svg License: CC BY-SA 3.0 Contributors: Own work Original artist: Geek3

• File:Arctangent_Arccotangent.svg Source: https://upload.wikimedia.org/wikipedia/commons/9/9a/Arctangent_Arccotangent.svg Li-cense: CC BY-SA 3.0 Contributors: Own work Original artist: Geek3

• File:Cercle_trigo.png Source: https://upload.wikimedia.org/wikipedia/commons/0/05/Cercle_trigo.png License: CC-BY-SA-3.0 Con-tributors: réalisé avec un programme de dessin vectoriel par Cdang Original artist: Christophe Dang Ngoc Chan Cdang at fr.wikipedia

• File:Compfun.svg Source: https://upload.wikimedia.org/wikipedia/commons/c/c9/Compfun.svg License: Public domain Contributors:Own work Original artist: Tlep

• File:Complex_ArcCot.jpg Source: https://upload.wikimedia.org/wikipedia/commons/6/60/Complex_ArcCot.jpg License: Public do-main Contributors: Eigenes Werk (own work) made with mathematica Original artist: Jan Homann

• File:Complex_ArcCsc.jpg Source: https://upload.wikimedia.org/wikipedia/commons/f/fd/Complex_ArcCsc.jpg License: Public do-main Contributors: Eigenes Werk (own work) made with mathematica Original artist: Jan Homann

• File:Complex_ArcSec.jpg Source: https://upload.wikimedia.org/wikipedia/commons/e/ec/Complex_ArcSec.jpg License: Public do-main Contributors: Eigenes Werk (own work) made with mathematica Original artist: Jan Homann

• File:Complex_arccos.jpg Source: https://upload.wikimedia.org/wikipedia/commons/4/4d/Complex_arccos.jpgLicense: Public domainContributors: made with mathematica, own work Original artist: Jan Homann

• File:Complex_arcsin.jpg Source: https://upload.wikimedia.org/wikipedia/commons/b/be/Complex_arcsin.jpg License: Public domainContributors: made with mathematica, own work Original artist: Jan Homann

• File:Complex_arctan.jpg Source: https://upload.wikimedia.org/wikipedia/commons/f/f5/Complex_arctan.jpg License: Public domainContributors: made with mathematica, own work Original artist: Jan Homann

• File:Composition_of_Inverses.png Source: https://upload.wikimedia.org/wikipedia/commons/4/4a/Composition_of_Inverses.png Li-cense: Public domain Contributors: Transferred from en.wikipedia to Commons. Original artist: Jim.belk at English Wikipedia

• File:Edit-clear.svg Source: https://upload.wikimedia.org/wikipedia/en/f/f2/Edit-clear.svg License: Public domain Contributors: TheTango! Desktop Project. Original artist:The people from the Tango! project. And according to themeta-data in the file, specifically: “Andreas Nilsson, and Jakub Steiner (althoughminimally).”

• File:Gràfica_del_arcsinus.png Source: https://upload.wikimedia.org/wikipedia/commons/6/66/Gr%C3%A0fica_del_arcsinus.pngLi-cense: GFDL Contributors: ? Original artist: ?

• File:Hyperbola_one_over_x.svg Source: https://upload.wikimedia.org/wikipedia/commons/4/43/Hyperbola_one_over_x.svg License:CC-BY-SA-3.0 Contributors: ? Original artist: ?

• File:Inversa_d'una_cúbica_gràfica.png Source: https://upload.wikimedia.org/wikipedia/commons/9/90/Inversa_d%27una_c%C3%BAbica_gr%C3%A0fica.png License: Public domain Contributors: Transferred from en.wikipedia Original artist: Original uploader wasJim.belk at en.wikipedia

• File:Inverse_Function.png Source: https://upload.wikimedia.org/wikipedia/commons/c/c8/Inverse_Function.png License: Public do-main Contributors: Transferred from en.wikipedia to Commons. Original artist: Jim.belk at English Wikipedia

• File:Inverse_Function_Graph.png Source: https://upload.wikimedia.org/wikipedia/commons/1/11/Inverse_Function_Graph.png Li-cense: Public domain Contributors: Transferred from en.wikipedia to Commons. Original artist: Jim.belk at English Wikipedia

• File:Inverse_Functions_Domain_and_Range.png Source: https://upload.wikimedia.org/wikipedia/commons/9/9d/Inverse_Functions_Domain_and_Range.png License: Public domain Contributors: Transferred from en.wikipedia to Commons. Original artist: Jim.belk atEnglish Wikipedia

• File:Inverse_square_graph.svg Source: https://upload.wikimedia.org/wikipedia/commons/7/70/Inverse_square_graph.svgLicense: CCBY-SA 3.0 Contributors: Own work Original artist: Krishnavedala

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• File:Trigonometric_functions_and_inverse2.svg Source: https://upload.wikimedia.org/wikipedia/commons/d/d8/Trigonometric_functions_and_inverse2.svg License: CC0 Contributors: Own work Original artist: Maschen

44 CHAPTER 4. MULTIPLICATIVE INVERSE

• File:Trigonometric_functions_and_inverse3.svg Source: https://upload.wikimedia.org/wikipedia/commons/f/fa/Trigonometric_functions_and_inverse3.svg License: CC0 Contributors: Own work Original artist: Maschen

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