PRECALCULUS

Inverse Relations and Functions

Inverse Relations and Functions

If two relations or functions are inverses, one relation

contains the point (x, y) and the other relation contains the point (y, x).

*Their graphs will be symmetric with respect to

the line y = x.

Sketch the inverse of the graph:

One-to-One / Horizontal Line Test

A function is one-to-one if no two x values have the same y value.

We can determine if a function is one-to-one by evaluating the graph of the original function using the Horizontal Line Test.

If a horizontal line passes through the graph in more than one point, the function is not

one-to-one, and the inverse of the function is not a function.

Determine whether the inverse of each function is a function.

1.

3. 4.

2.

Find the inverse of f(x) algebraically.

f ( x )= −1x−2

+4

Find the inverse of h(x) algebraically.

2( ) 2( 3) 6h x x

Given , what is the domain and range of

What is the domain and range of

=

Definition of an Inverse Function

Let f and g be two functions such that the following two conditions are met.

1. f(g(x))= x for every x in the domain of g

2. g(f(x))= x for every x in the domain of f

If these two conditions are met, then g(x) is the inverse of f(x) and is denoted .

Verify that f(x) and g(x) are inverse functions algebraically.

Given:

Verify that f(x) and g(x) are inverse functions algebraically.

Verify that f(x) and g(x) are inverse functions algebraically.

g

Which of the following statements is true?

A. The inverse of is .B. The function f(x) = 5 is one-to-one.C. If , then .D. The domain of f is the same as the range of

Extra Practice

Write the equation of a function that will have an inverse that is NOT a function.

Extra Practice

Write the equation of a function that will have an inverse that IS a function.

Extra Practice

Write the equation of a function, f, whose inverse has the range .

Extra Practice

Write the equation of a function, g, such that the inverse of g will have the domain .

Extra Practice

Verify that and are inverse functions.