# The condition for a function to have an inverse and find which function has an inverse from given two functions f ( x ) = x 2 and g ( x ) = x 3 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071 ### Precalculus: Mathematics for Calcu...

6th Edition
Stewart + 5 others
Publisher: Cengage Learning
ISBN: 9780840068071

#### Solutions

Chapter 2.7, Problem 2E

(a)

To determine

## To evaluate: The condition for a function to have an inverse and find which function has an inverse from given two functions f(x)=x2 and g(x)=x3

Expert Solution

For a function to have an inverse it must be one-to-one and function g(x)=x3 has an inverse.

### Explanation of Solution

Given:

The two functions are,

f(x)=x2 .                                                                                                                              ...... (1)

g(x)=x3 . (2)

Calculation:

There is a condition for a function to have an inverse which is that function must be one-to-one

From function (1),

A function is one-to-one if different inputs produce different outputs.

Substitute different values for x and find the outputs of function f(x)=x2 ,

 x f(x) 1 1 −1 1

From above table, the values of function f(1) and f(1) are same, so function f(x)=x2 is not a one-to-one function.

From function (2),

A function is one-to-one if different inputs produce different outputs.

Substitute different values for x and find the outputs of function g(x)=x3 ,

 x g(x) 1 1 −1 −1

From above table, the values of function g(1) and g(1) are  not same, so function f(x)=x2 is a one-to-one function

There is a condition for a function to have an inverse which is that function must be one-to-one

Thus, function g(x)=x3 has an inverse.

(b)

To determine

### To find: The inverse of a function g(x)=x3 .

Expert Solution

The inverse of the function g(x)=x3 is x13 .

### Explanation of Solution

Given:

The function is given below,

g(x)=x3 (3)

Calculation:

Let g(x) is assumed to be equal to y,

So, the function g(x) becomes,

y=g(x)

Substitute y for g(x) in equation (3), and find the value of x,

y=x3y13=x

Interchange the x and y in above equation,

y=x13

The above equation is an inverse function of g(x) ,

g1(x)=x13

Thus, the inverse of the function g(x)=x3 is x13 .

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