Chapter 1.2, Problem 51E

(a)

To determine

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

Answer to Problem 51E

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

Explanation of Solution

Given information: The functions are $f\left(x\right)=x+5$ and $g\left(x\right)=x^2-3$ .

Calculation:

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

  $f\left(g\left(x\right)\right)=f\left(x^2-3\right)$

As given $f\left(x\right)=x+5$ , substitute $x^2-3$ for $x$ in the function.

  $\begin{array}{c}f\left(x^2-3\right)=x^2-3+5\\ =x^2+2\end{array}$

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

(b)

To determine

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

Answer to Problem 51E

The value of composite function $g\left(f\left(x\right)\right)$ is $x^2+10x+22$ .

Explanation of Solution

Given information: The functions are $f\left(x\right)=x+5$ and $g\left(x\right)=x^2-3$ .

Calculation:

Substitute $x+5$ for $f\left(x\right)$ in the function $g\left(f\left(x\right)\right)$ .

  $g\left(f\left(x\right)\right)=g\left(x+5\right)$

As given $g\left(x\right)=x^2-3$ , substitute $x+5$ for $x$ in the function.

  $\begin{array}{c}g\left(x+5\right)={\left(x+5\right)}^2-3\\ =x^2+25+10x-3\\ =x^2+10x+22\end{array}$

Therefore, the value of composite function $g\left(f\left(x\right)\right)$ is $x^2+10x+22$ .

(c)

To determine

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

Answer to Problem 51E

The value of the function $f\left(g\left(0\right)\right)$ is $2$ .

Explanation of Solution

Given information: The functions are $f\left(x\right)=x+5$ and $g\left(x\right)=x^2-3$ .

Calculation:

As calculated in part (a), the function $f\left(g\left(x\right)\right)$ is $x^2+2$ .

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

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

Therefore, the value of the function $f\left(g\left(0\right)\right)$ is $2$ .

(d)

To determine

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

Answer to Problem 51E

The value of the function $g\left(f\left(0\right)\right)$ is $22$ .

Explanation of Solution

Given information: The functions are $f\left(x\right)=x+5$ and $g\left(x\right)=x^2-3$ .

Calculation:

As calculated in part (b), the value of composite function $g\left(f\left(x\right)\right)$ is $x^2+10x+22$ .

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

  $\begin{array}{c}g\left(f\left(0\right)\right)=0^2+10\times 0+22\\ g\left(f\left(0\right)\right)=22\end{array}$

Therefore, the value of the function $g\left(f\left(0\right)\right)$ is $22$ .

(e)

To determine

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

Answer to Problem 51E

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

Explanation of Solution

Given information: The function is $g\left(x\right)=x^2-3$ .

Calculation:

As the function is $g\left(x\right)=x^2-3$ , substitute $-2$ for $x$ in the function.

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

Substitute $1$ for $g\left(-2\right)$ in the function $g\left(g\left(-2\right)\right)$ .

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

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

(f)

To determine

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

Answer to Problem 51E

The value of the function $f\left(f\left(x\right)\right)$ is $x+10$ .

Explanation of Solution

Given information: The function is $f\left(x\right)=x+5$ .

Calculation:

Substitute $x+5$ for $f\left(x\right)$ in the function $f\left(f\left(x\right)\right)$ .

  $f\left(f\left(x\right)\right)=f\left(x+5\right)$

Substitute $x+5$ for $x$ in the given function.

  $\begin{array}{c}f\left(x+5\right)=\left(x+5\right)+5\\ =x+10\end{array}$

Therefore, the value of the function $f\left(f\left(x\right)\right)$ is $x+10$ .