Chapter 1.2, Problem 52E

(a)

To determine

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

Answer to Problem 52E

The value of composite function $f\left(g\left(x\right)\right)$ is $x$ .

Explanation of Solution

Given information: The functions are $f\left(x\right)=x+1$ and $g\left(x\right)=x-1$ .

Calculation:

Substitute $x-1$ for $g\left(x\right)$ in the function $f\left(g\left(x\right)\right)$ .

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

As it is given that $f\left(x\right)=x+1$ , substitute $x-1$ for $x$ in the function.

  $\begin{array}{c}f\left(x-1\right)=\left(x-1\right)+1\\ =x\end{array}$

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

(b)

To determine

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

Answer to Problem 52E

The value of composite function $g\left(f\left(x\right)\right)$ is $x$ .

Explanation of Solution

Given information: The functions are $f\left(x\right)=x+1$ and $g\left(x\right)=x-1$ .

Calculation:

Substitute $x+1$ for $f\left(x\right)$ in the function $g\left(f\left(x\right)\right)$ .

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

As it is given that $g\left(x\right)=x-1$ , substitute $x+1$ for $x$ in the function.

  $\begin{array}{c}g\left(x+1\right)=\left(x+1\right)-1\\ =x\end{array}$

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

(c)

To determine

To find: The value of the function $f\left(g\left(0\right)\right)$ .

Answer to Problem 52E

The value of the function $f\left(g\left(0\right)\right)$ is $0$ .

Explanation of Solution

Given information: The functions are $f\left(x\right)=x+1$ and $g\left(x\right)=x-1$ .

Calculation:

As calculated in part (a), the function $f\left(g\left(x\right)\right)$ is equal to $x$ .

Substitute $0$ for $x$ in the composite function $f\left(g\left(x\right)\right)$ .

  $\begin{array}{c}f\left(g\left(x\right)\right)=x\\ f\left(g\left(0\right)\right)=0\end{array}$

Therefore, the value of the function $f\left(g\left(0\right)\right)$ is $0$ .

(d)

To determine

To find: The value of the function $g\left(f\left(0\right)\right)$ .

Answer to Problem 52E

The value of the function $g\left(f\left(0\right)\right)$ is $0$ .

Explanation of Solution

Given information: The functions are $f\left(x\right)=x+1$ and $g\left(x\right)=x-1$ .

Calculation:

As calculated in part (b), the function $g\left(f\left(x\right)\right)$ is equal to $x$ .

Substitute $0$ for $x$ in the composite function $g\left(f\left(x\right)\right)$ .

  $\begin{array}{c}g\left(f\left(x\right)\right)=x\\ g\left(f\left(0\right)\right)=0\end{array}$

Therefore, the value of the function $g\left(f\left(0\right)\right)$ is $0$ .

(e)

To determine

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

Answer to Problem 52E

The value of the function $g\left(g\left(-2\right)\right)$ is $-4$ .

Explanation of Solution

Given information: The function is $g\left(x\right)=x-1$ .

Calculation:

As the function is $g\left(x\right)=x-1$ , substitute $-2$ for $x$ in the function.

  $\begin{array}{c}g\left(-2\right)=-2-1\\ =-3\end{array}$

Substitute $-3$ for $g\left(-2\right)$ in the function $g\left(g\left(-2\right)\right)$ .

  $\begin{array}{c}g\left(g\left(-2\right)\right)=g\left(-3\right)\\ =-3-1\\ =-4\end{array}$

Therefore, the value of the function $g\left(g\left(-2\right)\right)$ is $-4$ .

(f)

To determine

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

Answer to Problem 52E

The value of the function $f\left(f\left(x\right)\right)$ is $x+2$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=x+1$ .

Calculation:

Substitute $x+1$ for $f\left(x\right)$ in the function $f\left(f\left(x\right)\right)$ .

  $f\left(f\left(x\right)\right)=f\left(x+1\right)$

It is given that $f\left(x\right)=x+1$ , substitute $x+1$ for $x$ in the function.

  $\begin{array}{c}f\left(x+1\right)=\left(x+1\right)+1\\ =x+2\end{array}$

Therefore, the value of the function $f\left(f\left(x\right)\right)$ is $x+2$ .