Chapter 1, Problem 42RE

(a)

To determine

To find: The value of the function $\left(f\circ g\right)\left(-1\right)$ .

Answer to Problem 42RE

The value of the function $\left(f\circ g\right)\left(-1\right)$ is $1$ .

Explanation of Solution

Given information: The given functions are $f\left(x\right)=2-x$ and $g\left(x\right)=\sqrt[3]{x+1}$ .

Calculation:

The composite function $\left(f\circ g\right)\left(-1\right)$ can be written as:

  $\left(f\circ g\right)\left(-1\right)=f\left(g\left(-1\right)\right)$

Substitute $-1$ for $x$ in the function $g\left(x\right)=\sqrt[3]{x+1}$ .

  $\begin{array}{c}g\left(1\right)=\sqrt[3]{\left(-1\right)+1}\\ =0\end{array}$

Now, substitute 0 for $x$ in the function $f\left(x\right)=2-x$ .

  $\begin{array}{c}f\left(0\right)=2-0\\ =2\end{array}$

Therefore, the value of the function $\left(f\circ g\right)\left(-1\right)$ is 1.

(b)

To determine

To find: The value of the function $\left(g\circ f\right)\left(2\right)$ .

Answer to Problem 42RE

The value of the function $\left(g\circ f\right)\left(2\right)$ is 1.

Explanation of Solution

Given information: The given functions are $f\left(x\right)=2-x$ and $g\left(x\right)=\sqrt[3]{x+1}$ .

Calculation:

The composite function $\left(g\circ f\right)\left(2\right)$ can be written as:

  $\left(g\circ f\right)\left(2\right)=g\left(f\left(2\right)\right)$

Substitute 2 for $x$ in the function $f\left(x\right)=2-x$ .

  $\begin{array}{c}f\left(2\right)=2-2\\ =0\end{array}$

Now, substitute 0 for $x$ in the function $g\left(x\right)=\sqrt[3]{x+1}$ .

  $\begin{array}{c}g\left(0\right)=\sqrt[3]{0+1}\\ =1\end{array}$

Therefore, the value of the function $\left(g\circ f\right)\left(2\right)$ is 1.

(c)

To determine

To find: The value of the function $\left(f\circ f\right)\left(x\right)$ .

Answer to Problem 42RE

The value of the function $\left(f\circ f\right)\left(x\right)$ is $x$ .

Explanation of Solution

Given information: The given function is $f\left(x\right)=2-x$ .

Calculation:

The composite function $\left(f\circ f\right)\left(x\right)$ can be calculated as:

  $\begin{array}{c}\left(f\circ f\right)\left(x\right)=f\left(f\left(x\right)\right)\\ =f\left(2-x\right)\\ =2-\left(2-x\right)\\ =x\end{array}$

Since, $f\left(x\right)=\frac{1}{x}$ has domain $x\ne 0$ .

Therefore, the value of the function $\left(f\circ f\right)\left(x\right)$ is $x$ .

(d)

To determine

To find: The value of function $\left(g\circ g\right)\left(x\right)$ .

Answer to Problem 42RE

The value of the function $\left(g\circ g\right)\left(x\right)$ is $\sqrt[3]{\left(\sqrt[3]{x+1}\right)+1}$ .

Explanation of Solution

Given information: The given function is $g\left(x\right)=\sqrt[3]{x+1}$ .

Calculation:

The function $\left(g\circ g\right)\left(x\right)$ can be calculated as:

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

Therefore, the value of the function $\left(g\circ g\right)\left(x\right)$ is $\sqrt[3]{\left(\sqrt[3]{x+1}\right)+1}$ .