Chapter 5, Problem 1RE

(a)

To determine

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

Explanation of Solution

Given: The functions are $f\left(x\right)=3x-5$ and $g\left(x\right)=1-2x^2$ .

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

  $\begin{array}{c}\left(f\circ g\right)\left(2\right)=f\left(g\left(2\right)\right)\\ =f\left(-7\right)\\ =-7\times 3-5\\ =-26\end{array}$

Therefore, the value of $\left(f\circ g\right)\left(2\right)$ is $-26$ .

(b)

To determine

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

Explanation of Solution

Given: The functions are $f\left(x\right)=3x-5$ and $g\left(x\right)=1-2x^2$ .

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

  $\begin{array}{c}\left(g\circ f\right)\left(-2\right)=g\left(f\left(-2\right)\right)\\ =g\left(-11\right)\\ =1-242\\ =-241\end{array}$

Therefore, the value of $\left(g\circ f\right)\left(-2\right)$ is $-241$ .

(c)

To determine

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

Explanation of Solution

Given: The functions are $f\left(x\right)=3x-5$ and $g\left(x\right)=1-2x^2$ .

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

  $\begin{array}{c}\left(f\circ f\right)\left(4\right)=f\left(f\left(4\right)\right)\\ =3\times 7-5\\ =16\end{array}$

Therefore, the value of $\left(f\circ f\right)\left(4\right)$ is $16$ .

(d)

To determine

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

Explanation of Solution

Given: The functions are $f\left(x\right)=3x-5$ and $g\left(x\right)=1-2x^2$ .

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

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

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