Section 4.1

Inverse Functions

What are Inverse Operations?

Inverse operations are operations that “undo” each other.

ExamplesAddition and Subtraction are inverse

operationsMultiplication and Division are inverse

operations

How to Find Inverse Relation

• Interchange coordinates of each ordered pair in the relation.

• If a relation is defined by an equation, interchange the variables.

Inverses• When we go from an output of a function

back to its input or inputs, we get an inverse relation. When that relation is a function, we have an inverse function.

• Interchanging the first and second coordinates of each ordered pair in a relation produces the inverse relation.

Example: Consider the relation g given by

g = {(2, 4), (3, 4), (8, 5). Solution: The inverse of the relation is

Inverse Relation

Inverse Relation• If a relation is defined by an equation,

interchanging the variables produces an equation of the inverse relation.

Example: Find an equation for the inverse of the relation:

y = x2 2x.

Solution: We interchange x and y and obtain an equation of the inverse:

• Graphs of a relation and its inverse are always reflections of each other across the line y = x.

Inverse Relation

One-to-One Functions

4

3

2

1

5

6

7

8

f

4

3

2

1

6

8

9

g

f is one-to-one g is not one-to-one

One-to-One Function

A function f is a one-to-one function if,

for two elements a and b from the domain

of f,

a b implies f(a) f(b)

** x-values do not share the same y-value **

Horizontal Line Test

• Use to determine whether a function is one-to-one

• A function is one-to-one if and only if no horizontal line intersects its graph more than once.

Horizontal-Line Test

Graph f(x) = 3x + 4. Example: From the graph at the left, determine whether the function is one-to-one and thus has an inverse that is a function.

Solution: No horizontal line intersects the graph more than once, so the function is one-to-one. It has an inverse that is a function.

f(x) = 3x + 4

Horizontal-Line Test

Graph f(x) = x2 2. Example: From the

graph at the left, determine whether the function is one-to-one and thus has an inverse that is a function.

Solution: There are many horizontal lines that intersect the graph more than once. The inputs 1 and 1 have the same output, 1. Thus the function is not one-to-one. The inverse is not a function.

Why are one-to-one functions important?

One-to-One Functions have

Inverse functions

Inverses of Functions

If the inverse of a function f is also a function, it is named f 1 and read “f-inverse.”

The negative 1 in f 1 is not an exponent. This does not mean the reciprocal of f.

f 1(x) is not equal to1

( )f x

One-to-One Functions

A function f is one-to-one if different inputs have different outputs.

That is, if a b then f(a) f(b).

A function f is one-to-one if when the outputs are the same, the inputs are the same.

That is, if f(a) = f(b) then a = b.

Properties of One-to-One Functions and Inverses

• If a function is one-to-one, then its inverse is a function.

• The domain of a one-to-one function f is the range of the inverse f 1.

• The range of a one-to-one function f is the domain of the inverse f 1.

• A function that is increasing over its domain or is decreasing over its domain is a one-to-one function.

Inverse Functions

x Y = f(x)

A Bf

f-1

Domain of f-1 = range of f

Range of f-1 = domain of f

How to find the Inverse of a One-to-One Function

1. Replace f(x) with y in the equation.

2. Interchange x and y in the equation.

3. Solve this equation for y.

4. Replace y with f-1(x).

Any restrictions on x or y should be considered.Remember: Domain and Range are interchanged

for inverses.

Example

Determine whether the function f(x) = 3x 2 is one-to-one, and if it is, find a formula for f 1(x).

Solution: The graph is that of a line and passes the horizontal-line

test. Thus it is one-to-one and its inverse is a function.

1. Replace f(x) with y:

2. Interchange x and y:

3. Solve for y: 4. Replace y with f 1(x):

Graph of Inverse f-1 function

• The graph of f-1 is obtained by reflecting the graph of f across the line y = x.

• To graph the inverse f-1 function:Interchange the points on the graph of f to obtain the points on the graph of f-1. If (a,b) lies on f, then (b,a) lies on f-1.

Example

Graph f(x) = 3x 2 and f 1(x) = using the same set of axes.

Then compare the two graphs.

2

3

x

Solution

73

42

20

51

f(x) = 3x 2x

24

11

02

15

x

+ 2

3

xf 1(x) =

Inverse Functions and Composition

If a function f is one-to-one, then f 1 is the unique function such that each of the following holds:

for each x in the domain of f, and

for each x in the domain of f 1.

1 1

1 1

( )( ) ( ( ))

( )( ) ( ( ))

f f x f f x x

f f x f f x x

Example Given that f(x) = 7x 2, use composition of

functions to show that f 1(x) = (x + 2)/7.

Solution: 1 1

1

( )( ) ( ( ))

(7 2)

(7 2) 2

77

7

f f x f f x

f x

x

x

x

1 1( )( ) ( ( ))

2( )

72

7( ) 272 2

f f x f f x

xf

x

x

x

Restricting a Domain

• When the inverse of a function is not a function, the domain of the function can be restricted to allow the inverse to be a function.

• In such cases, it is convenient to consider “part” of the function by restricting the domain of f(x). If the domain is restricted, then its inverse is a function.

Restricting the Domain

Recall that if a function is not one-to-one, then its inverse will not be a function.

Restricting the Domain

If we restrict the domain values of f(x) to those greater than or equal to zero, we see that f(x) is now one-to-one and its inverse is now a function.