Chapter 5, Problem 7RE

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 $f\left(x\right)=2-x$ and $g\left(x\right)=3x+1$ .

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

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

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,

  $\begin{array}{c}\left(g\circ f\right)\left(c\right)=3\left(2-x\right)+1\\ =6-3x+1\\ =7-3x\end{array}$

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,

  $\begin{array}{c}\left(f\circ f\right)\left(x\right)=2-\left(2-x\right)\\ =2-2+x\\ =x\end{array}$

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,

  $\begin{array}{c}\left(g\circ g\right)\left(x\right)=3\left(3x+1\right)+1\\ =9x+3+1\\ =9x+4\end{array}$

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