Chapter 5, Problem 3RE

(a)

To determine

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

Explanation of Solution

Given: The functions are $f\left(x\right)=\sqrt{x+2}$ and $g\left(x\right)=2x^2+1$ .

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

  $\begin{array}{c}\left(f\circ g\right)\left(2\right)=f\left(g\left(2\right)\right)\\ =f\left(9\right)\\ =\sqrt{9+2}\\ =\sqrt{11}\end{array}$

Therefore, the value of $\left(f\circ g\right)\left(2\right)$ is $\sqrt{11}$ .

(b)

To determine

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

Explanation of Solution

Given: The functions are $f\left(x\right)=\sqrt{x+2}$ and $g\left(x\right)=2x^2+1$ .

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

  $\begin{array}{c}\left(g\circ f\right)\left(-2\right)=g\left(f\left(-2\right)\right)\\ =g\left(0\right)\\ =2{\left(0\right)}^2+1\\ =1\end{array}$

Therefore, the value of $\left(g\circ f\right)\left(-2\right)$ is $1$ .

(c)

To determine

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

Explanation of Solution

Given: The functions are $f\left(x\right)=\sqrt{x+2}$ and $g\left(x\right)=2x^2+1$ .

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

  $\begin{array}{c}\left(f\circ f\right)\left(4\right)=f\left(f\left(4\right)\right)\\ =f\left(\sqrt{6}\right)\\ =\sqrt{\sqrt{6}+2}\end{array}$

Therefore, the value of $\left(f\circ f\right)\left(4\right)$ is $\sqrt{\sqrt{6}+2}$ .

(d)

To determine

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

Explanation of Solution

Given: The functions are $f\left(x\right)=\sqrt{x+2}$ and $g\left(x\right)=2x^2+1$ .

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

  $\begin{array}{c}\left(g\circ g\right)\left(-1\right)=g\left(g\left(-1\right)\right)\\ =g\left(3\right)\\ =2{\left(3\right)}^2+1\\ =19\end{array}$

Therefore, the value of $\left(g\circ g\right)\left(-1\right)$ is $19$ .