Chapter 5, Problem 9RE

To determine

To find : The expression for $f\circ g$ , $f\circ f$ , $g\circ f$ , $g\circ g$ and state the domain of each composite function.

Explanation of Solution

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

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

  $\begin{array}{c}\left(f\circ g\right)\left(x\right)=3{\left(\left|3x\right|\right)}^2+\left|3x\right|+1\\ =27x^2+3x+1\end{array}$

The domain of $g$ is the set of all real numbers.

So, the domain of $f\circ g$ is the set of all real numbers.

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

  $\begin{array}{c}\left(g\circ f\right)\left(x\right)=\left|3\left(3x^2+x-+\right)\right|\\ =\left|9x^2+3x+3\right|\end{array}$

The domain of $f$ is the set of all real numbers.

So, the domain of $g\circ f$ is the set of all real numbers

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

  $\begin{array}{c}\left(f\circ f\right)\left(x\right)=3{\left(3x^2+x+1\right)}^2+3x^2+x+1+1\\ =27x^4+18x^3+24x^2+7x+5\end{array}$

The domain of $f$ is the set of all real numbers.

So, the domain of $\left(f\circ f\right)\left(x\right)$ is the set of all real numbers

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

  $\begin{array}{c}\left(g\circ g\right)\left(x\right)=\left|3\left|3x\right|\right|\\ =\left|9x\right|\end{array}$

The domain of $g$ is the set of all real numbers.

So, the domain of $\left(g\circ g\right)\left(x\right)$ is the set of all real numbers