Chapter 5, Problem 8RE

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

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

  $\begin{array}{c}\left(f\circ g\right)\left(x\right)=2\left(2x+1\right)-1\\ =4x+1\end{array}$

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,

  $\begin{array}{c}\left(g\circ f\right)\left(x\right)=2\left(2x-1\right)+1\\ =4x-1\end{array}$

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,

  $\begin{array}{c}\left(f\circ f\right)\left(x\right)=2\left(2x-1\right)-1\\ =4x-3\end{array}$

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,

  $\begin{array}{c}\left(g\circ g\right)\left(x\right)=2\left(2x+1\right)+1\\ =4x+2\end{array}$

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