Section 6.1 Section 6.2 Composite Functions Inverse Functions.
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Transcript of Section 6.1 Section 6.2 Composite Functions Inverse Functions.
Section 6.1Section 6.2
Composite Functions
Inverse Functions
THE COMPOSITE FUNCTION
Given two function f and g, the composite function, denote by f ◦ g (read “f composed with g”), is defined by
( f ◦ g)(x) = f (g(x))
The domain of f ◦ g is the set of all numbers x in the domain of g such that g(x) is in the domain of f.
CONCEPT OF AN INVERSE FUNCTION
Idea: An inverse function takes the output of the “original” function and tells from what input it resulted.
Note that this really says that the roles of x and y are reversed.
MATHEMATICAL DEFINITION OF INVERSE FUNCTIONS
1. ( )( )f g x x x D g fo r a ll in
In the language of function notation, two functions f and g are inverses of each other if and only if
2. ( )( )g f x x x D f fo r a ll in
NOTATION FOR THE INVERSE FUNCTION
f x 1 ( )
f x 1 ( )
We use the notation
for the inverse of f(x).
NOTE: does NOT mean1
f x( )
ONE-TO-ONE FUNCTIONS
A function is one-to-one if for each y-value there is only one x‑value that can be paired with it; that is, each output comes from only one input.
ONE-TO-ONE FUNCTIONS AND INVERSE FUNCTIONS
Theorem: A function has an inverse if and only if it is one-to-one.
TESTING FOR AONE-TO-ONE FUNCTION
Horizontal Line Test: A function is one-to-one (and has an inverse) if and only if no horizontal line touches its graph more than once.
GRAPHING ANINVERSE FUNCTION
Given the graph of a one-to-one function, the graph of its inverse is obtained by switching x- and y-coordinates.
The resulting graph is reflected about the line y = x.
FINDING A FORMULA FORAN INVERSE FUNCTION
To find a formula for the inverse given an equation for a one-to-one function:
1. Replace f (x) with y.
2. Interchange x and y.
3. Solve the resulting equation for y.
4. Replace y with f -1(x).