Chapter 4, Problem 1RE

(a)

To determine

The value of $\left(g\circ f\right)\left(-8\right)$ by using the graph of $y=f\left(x\right)$ and $y=g\left(x\right)$ from the given figure.

Answer to Problem 1RE

The value of the function $\left(g\circ f\right)\left(x\right)$ at $x=-8$ is $\underset{\_}{-4}$.

Explanation of Solution

Definition used:

Given two functions f and g, the composite function, denoted by $f\circ g$ (read as “f composed with g”), is defined by $\left(f\circ g\right)\left(x\right)=f\left(g\left(x\right)\right)$.

Calculation:

It is observed from the figure that the value of $f\left(x\right)$ at $x=-8$ is $f\left(-8\right)=2$ and the value of $g\left(x\right)$ at $x=2$ is $g\left(2\right)=-4$.

By using the above definition $\left(g\circ f\right)\left(-8\right)$ can be calculated as follows.

$\begin{array}{c}\left(g\circ f\right)\left(-8\right)=g\left(f\left(-8\right)\right)\\ =g\left(2\right)\\ =-4\end{array}$

Thus, the value of the function $\left(g\circ f\right)\left(x\right)$ at $x=-8$ is $\underset{\_}{-4}$.

(b)

To determine

The value of $\left(f\circ g\right)\left(-8\right)$ by using the graph of $y=f\left(x\right)$ and $y=g\left(x\right)$ from the given figure.

Answer to Problem 1RE

The value of the function $\left(f\circ g\right)\left(x\right)$ at $x=-8$ is $\underset{\_}{1}$.

Explanation of Solution

It is observed from the figure that the value of value of $g\left(x\right)$ at $x=-8$ is $g\left(-8\right)=-2$ and the value of $f\left(x\right)$ at $x=-2$ is $f\left(-2\right)=1$.

By using the above definition $\left(f\circ g\right)\left(-8\right)$ can be calculated as follows.

$\begin{array}{c}\left(f\circ g\right)\left(-8\right)=f\left(g\left(-8\right)\right)\\ =f\left(-2\right)\\ =1\end{array}$

Thus, the value of the function $\left(f\circ g\right)\left(x\right)$ at $x=-8$ is $\underset{\_}{1}$.

(c)

To determine

The value of $\left(g\circ g\right)\left(7\right)$ by using the graph of $y=f\left(x\right)$ and $y=g\left(x\right)$ from the given figure.

Answer to Problem 1RE

The value of the function $\left(g\circ g\right)\left(x\right)$ at $x=7$ is $\underset{\_}{-6}$.

Explanation of Solution

It is observed from the figure that the value of $g\left(x\right)$ at $x=7$ is $g\left(7\right)=0$ and the value of $g\left(x\right)$ at $x=0$ is $g\left(0\right)=-6$.

By using the above definition $\left(g\circ g\right)\left(7\right)$ can be calculated as follows.

$\begin{array}{c}\left(g\circ g\right)\left(7\right)=g\left(g\left(7\right)\right)\\ =g\left(0\right)\\ =-6\end{array}$

Thus, the value of the function $\left(g\circ g\right)\left(x\right)$ at $x=7$ is $\underset{\_}{-6}$.

(d)

To determine

The value of $\left(g\circ f\right)\left(-5\right)$ by using the graph of $y=f\left(x\right)$ and $y=g\left(x\right)$ from the given figure.

Answer to Problem 1RE

The value of the function $\left(g\circ f\right)\left(x\right)$ at $x=-5$ is $\underset{\_}{-6}$.

Explanation of Solution

It is observed from the figure that the value of $f\left(x\right)$ at $x=-5$ is $f\left(-5\right)=4$ and the value of $g\left(x\right)$ at $x=4$ is $g\left(4\right)=-6$.

By using the above definition $\left(g\circ f\right)\left(-5\right)$ can be calculated as follows.

$\begin{array}{c}\left(g\circ f\right)\left(-5\right)=g\left(f\left(-5\right)\right)\\ =g\left(4\right)\\ =-6\end{array}$

Thus, the value of the function $\left(g\circ f\right)\left(x\right)$ at $x=-5$ is $\underset{\_}{-6}$.