College Algebra Chapter 4 Exponential and Logarithmic Functions Section 4.1 Inverse Functions Copyright © 2017 McGraw-Hill Education. All rights reserved. No reproduction or distribution without the prior written consent of McGraw-Hill Education.

Concepts Identify One-to-One Functions Determine Whether Two Functions Are Inverses Find the Inverse of a Function

Concept 1 Identify One-to-One Functions

Identify One-to-One Functions One-to-One Function: A function f is a one-to-one function if for a and b in the domain of f, if a ≠ b, then f(a) ≠ f(b), or equivalently, if f(a) = f(b), then a = b. A function y = f (x) is a one-to-one function if no horizontal line intersects the graph in more than one place.

Example 1 Determine if the relation defines y as a one-to-one function of x. {(2,1), (4,2), (7,3), (-2,1)}

Example 2 Determine if the relation defines y as a one-to- one function of x.

Skill Practice 1 Determine whether the function is one-to-one.

Example 3 Determine if the relation defines y as a one-to-one function of x.

Example 4 Determine if the relation defines y as a one-to-one function of x.

Example 5 Determine if the relation defines y as a one-to-one function of x.

Skill Practice 2 Use the horizontal line test to determine if the graph defines y as a one-to-one function of x.

Example 6 Use the definition of a one-to-one function to determine whether the function is one-to-one. (Show that if f(a) = f(b) then a = b) f(x) = 3x + 2

Example 7 Use the definition of a one-to-one function to determine whether the function is one-to-one. (Show that if f(a) = f(b) then a = b

Skill Practice 3 Determine whether the function is one-to-one. f(x) = -4x + 1 f(x) = |x| - 3

Concept 2 Determine Whether Two Functions Are Inverses

Determine Whether Two Functions Are Inverses Inverse Functions: Let f be a one-to-one function. Then g is the inverse of f if the following conditions are both true. Given a function f(x) and its inverse then the definition implies that

Example 8 Determine whether the two functions are inverses.

Example 9 Determine whether the two functions are inverses.

Skill Practice 4 Determine whether the function are inverses.

Concept 3 Find the Inverse of a Function

Find the Inverse of a Function Procedure to Find an Equation of an Inverse of a Function For a one-to-one function defined by y = f(x), the equation of the inverse can be found as follows: Step 1: Replace f(x) by y. Step 2: Interchange x and y. Step 3: Solve for y. Step 4: Replace y by

Example 10 A one-to-one function is given. Write an equation for the inverse function.

Skill Practice 5 Write an equation for the inverse function for f(x) = 4x + 3.

Example 11 A one-to-one function is given. Write an equation for the inverse function.

Skill Practice 6 Write an equation for the inverse function for the one-to-one function defined by

Example 12 A one-to-one function is given. Write an equation for the inverse function.

Skill Practice 7 write an equation of the inverse.

Skill Practice 8 find an equation of the inverse.

Example 13 The graph of a function is given. Graph the inverse function.