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).