Chapter 6, Problem 1RE

(a)

To determine

The composite function $\left(g\circ f\right)\left(-8\right)$.

Answer to Problem 1RE

The composite function $\underset{\_}{\left(g\circ f\right)\left(-8\right)=-4}$.

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:

From the given graph, it can be observed that, $f\left(-8\right)=2$.

Using the above definition, obtain the composite function $\left(g\circ f\right)\left(-8\right)$ as shown below:

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

Thus, the composite function $\underset{\_}{\left(g\circ f\right)\left(-8\right)=-4}$.

(b)

To determine

The composition function $\left(f\circ g\right)\left(-8\right)$.

Answer to Problem 1RE

The composite function $\underset{\_}{\left(f\circ g\right)\left(-8\right)=1}$.

Explanation of Solution

From the given graph, it can be observed that, $g\left(-8\right)=-2$.

Using the above definition, obtain the composite function $\left(f\circ g\right)\left(-8\right)$ as shown below:

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

Thus, the composite function $\underset{\_}{\left(f\circ g\right)\left(-8\right)=1}$.

(c)

To determine

The composition function $\left(g\circ g\right)\left(7\right)$.

Answer to Problem 1RE

The composite function $\underset{\_}{\left(g\circ g\right)\left(7\right)=-6}$.

Explanation of Solution

From the given graph, it can be observed that, $g\left(7\right)=0$.

Using the above definition, obtain the composite function $\left(g\circ g\right)\left(7\right)$ as shown below:

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

Thus, the composite function $\underset{\_}{\left(g\circ g\right)\left(7\right)=-6}$.

(d)

To determine

The composite function $\left(g\circ f\right)\left(-5\right)$.

Answer to Problem 1RE

The composite function $\underset{\_}{\left(g\circ f\right)\left(-5\right)=-6}$.

Explanation of Solution

From the given graph, it can be observed that, $f\left(-5\right)=4$.

Using the above definition, obtain the composite function $\left(g\circ f\right)\left(-5\right)$ as shown below:

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

Thus, the composite function $\underset{\_}{\left(g\circ f\right)\left(-5\right)=-6}$.