Chapter 5, Problem 5RE

(a)

To determine

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

Explanation of Solution

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

The value of $\left(f\circ g\right)\left(2\right)$ is calculated as,

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

Therefore, the value of $\left(f\circ g\right)\left(2\right)$ is $e^4$ .

(b)

To determine

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

Explanation of Solution

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

The value of $\left(g\circ f\right)\left(-2\right)$ is calculated as,

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

Therefore, the value of $\left(g\circ f\right)\left(-2\right)$ is $3e^{-2}-2$ .

(c)

To determine

To find : The value of $\left(f\circ f\right)\left(4\right)$

Explanation of Solution

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

The value of $\left(f\circ f\right)\left(4\right)$ is calculated as,

  $\begin{array}{c}\left(f\circ f\right)\left(4\right)=f\left(f\left(4\right)\right)\\ =f\left(e^4\right)\\ =e^{e^4}\end{array}$

Therefore, the value of $\left(f\circ f\right)\left(4\right)$ is $e^{e^4}$ .

(d)

To determine

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

Explanation of Solution

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

The value of $\left(g\circ g\right)\left(-1\right)$ is calculated as,

  $\begin{array}{c}\left(g\circ g\right)\left(-1\right)=g\left(g\left(-1\right)\right)\\ =g\left(-5\right)\\ =-15-2\\ =-17\end{array}$

Therefore, the value of $\left(g\circ g\right)\left(-1\right)$ is $-17$ .