Chapter 7.6, Problem 65PPS

a.

To determine

Find the value of functions $f\left(x\right)=x^3\text{ and }g\left(x\right)=x^{\frac{1}{3}}$ at the points $-2,-1,0,1,2$ .

Answer to Problem 65PPS

The complete table is shown as,

At point (x)	f(x)=x3	g(x)=x1/3
-2	-8	(-2)1/3
-1	-1	-1
0	0	0
1	1	1
2	8	(2)1/3

Explanation of Solution

Given information: The given functions are $f\left(x\right)=x^3\text{ and }g\left(x\right)=x^{\frac{1}{3}}$ .

Calculation: The value of functions $f\left(x\right)=x^3\text{ and }g\left(x\right)=x^{\frac{1}{3}}$ at points $-2,-1,0,1,2$ is evaluated as,

At point (x)	f(x)=x3	g(x)=x1/3
-2	-8	(-2)1/3
-1	-1	-1
0	0	0
1	1	1
2	8	(2)1/3

b.

To determine

Sketch the graph of the functions $f\left(x\right)=x^3\text{and }g\left(x\right)=x^{\frac{1}{3}}$ .

Explanation of Solution

Given information: The given functions are $f\left(x\right)=x^3\text{and }g\left(x\right)=x^{\frac{1}{3}}$ .

Graph: The graphs for the functions $f\left(x\right)=x^3\text{and }g\left(x\right)=x^{\frac{1}{3}}$ is shown as,

  

c.

To determine

Explain the transformation between the functions $f\left(x\right)=x^3\text{ and }g\left(x\right)=x^{\frac{1}{3}}$ .

Answer to Problem 65PPS

The functions are inverse functions to each other.

Explanation of Solution

Given information: The given functions are $f\left(x\right)=x^3\text{ and }g\left(x\right)=x^{\frac{1}{3}}$ .

Calculation: The functions $f\left(x\right)=x^3\text{ and }g\left(x\right)=x^{\frac{1}{3}}$ are inverse functions to each other because $f\left(g\left(x\right)\right)=g\left(f\left(x\right)\right)$ . And both are equal to identity function.

Hence the transformation of given function is inverse.