Chapter 1.5, Problem 42E

(a)

To determine

To calculate: To determine $f\circ g$

Answer to Problem 42E

  $f\circ g=x$

Explanation of Solution

Given:

  $f(x)=\sqrt[3]{x-1}$

  $g(x)=x^3+1$

Formula Used:

Composition of functions is when one function is inside of another function.

The notation used for the composition of functions looks like this, $\left(f\circ g\right)\left(x\right)$ .

The composition of the function f with g is defined as follows:

  $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ , notice that in the case the function g is inside of the function

Calculation:

  $f\circ g$ is calculated as:

  $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$

Substitute: $g(x)=x^3+1$

  $\left(f\circ g\right)\left(x\right)=f\left(x^3+1\right)$

Also, given $f(x)=\sqrt[3]{x-1}$

Thus, $\left(f\circ g\right)\left(x\right)$ is calculated as

  $\begin{array}{l}\left(f\circ g\right)\left(x\right)=\sqrt[3]{x^3+1-1}\\ \left(f\circ g\right)\left(x\right)=\sqrt[3]{x^3}\\ \left(f\circ g\right)\left(x\right)=x\end{array}$

Conclusion:

Hence, $\left(f\circ g\right)\left(x\right)=x$

(b)

To determine

To calculate: To determine $g\circ f$

Answer to Problem 42E

  $g\circ f=x$

Explanation of Solution

Given:

  $f(x)=\sqrt[3]{x-1}$

  $g(x)=x^3+1$

Formula Used:

Composition of functions is when one function is inside of another function.

The notation used for the composition of functions looks like this, $\left(f\circ g\right)\left(x\right)$ .

The composition of the function f with g is defined as follows:

  $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ , notice that in the case the function g is inside of the function

Calculation:

Equation of $g\circ f$ is calculated as:

  $\left(g\circ f\right)\left(x\right)=g\left(f\left(x\right)\right)$

Substitute: $f(x)=\sqrt[3]{x-1}$

Thus,

  $\left(g\circ f\right)\left(x\right)=g\left(\sqrt[3]{x-1}\right)$

Also given that $g(x)=x^3+1$

Thus, $g\circ f$ is calculated as follows:

  $\begin{array}{l}\left(g\circ f\right)\left(x\right)={(}^{\sqrt[3]{x-1}}+1\\ \left(g\circ f\right)\left(x\right)=(x-1)+1\\ \left(g\circ f\right)\left(x\right)=x\end{array}$

Conclusion:

Hence, $\left(g\circ f\right)\left(x\right)=x$

(c)

To determine

To calculate: To determine $\left(f\circ g\right)\left(0\right)$

Answer to Problem 42E

  $\left(f\circ g\right)\left(0\right)=0$

Explanation of Solution

Given:

  $f(x)=\sqrt[3]{x-1}$

  $g(x)=x^3+1$

Formula Used:

Composition of functions is when one function is inside of another function.

The notation used for the composition of functions looks like this, $\left(f\circ g\right)\left(x\right)$ .

The composition of the function f with g is defined as follows: $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$ , notice that in the case the function g is inside of the function

Calculation:

From part (a):

  $\left(f\circ g\right)\left(x\right)=x$

To calculate $\left(f\circ g\right)\left(0\right)$ replace $x$ with 0:

  $\left(f\circ g\right)\left(0\right)=0$

Conclusion:

Hence, $\left(f\circ g\right)\left(0\right)=0$