e - book for college algebra king fahd university of petroleum & minerals 2.7 e - book for...

E - BOOK FOR COLLEGE ALGEBRA King Fahd University of Petroleum & Minerals

KFUPM - Prep Year Math Program (c) 20013 All Right Reserved

2.7


Inverse of Functions

Inverse Functions Properties of Inverse Functions Finding the Inverse Function



Inverse of Functions

Two functions and are said to be inverse of one another if the following two conditions are satisfied

, for all in , for all in



Example 1Verify that and are inverse of each other.

Since both and have domains all real numbers, we need to show that the above two conditions (i) and (ii) are satisfied for all real numbers. We have: and therefore, and g satisfy conditions (i) and (ii), hence and g are inverse functions.



Properties of Inverse Functions

Property 1: (follows from condition i) (follows from condition ii)

Property 2: A point lies on the graph of if and only if the point lies on the graph of .Property 3: The graph of is a reflection of the graph of about the line .Property 4:If the graph of two function f and g are reflection of one another through the line then the two functions are inverses to one another



Example 2For assuming that exists, find

Since , let



Example 3Assume that a function has an inverse and , find the value of for which .

We have

and applying to both sides of the equation gives which gives

or



Example 4Assuming that has inverse , find the formula for

Using property 2, . Therefore, all what we need to do is to solve the equation for , or for.



One-to-one Functions

A function with domain is said to be one-to-one if and only if wherever for some and in then .

For example, the functions: , , and ( is even) are all none one-to-one functions. While the functions: , , and ( is odd) are all one-to-one functions.



Example 5Determine which of the functions whose graphs are given below has inverse incase the function has an inverse give the graph of its inverse

𝑓 (𝑥 )=32𝑥−3 𝑔 (𝑥 )=2 −

12𝑥2 𝑘 (𝑥 )=√2𝑥



Example 5 Solution

The functions and are one-to-one functions because their graphs satisfy the horizontal line test, while the function is not one-to-one because its graph meets the horizontal lines , twice. The graphs of inverses of and are given by

𝑓 −1 (𝑥 ) 𝑘−1 (𝑥 )

𝑓 (𝑥 ) 𝑘 (𝑥 )

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