Chapter 6.2, Problem 4E

To determine

To plot: The graph of the given function $f\left(x\right)$ , its inverse $g\left(x\right)$ and $f\circ g\left(x\right)$ .

Answer to Problem 4E

The graph of the functions are given below.

Explanation of Solution

Given:

The given function is:

  $f\left(x\right)=\frac{1}{2}x+3\dots \left(1\right)$

The graph of the function $f\left(x\right)$ is given as:

  

Define the inverse of the given function is as:

  $g\left(x\right)=2x-6$

Check:

To check the above function is the inverse of the given function the calculation is given as:

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

Further simplified as:

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

As,

  $\begin{array}{l}f\circ g\left(x\right)=x\\ g\circ f\left(x\right)=x\end{array}$

So, defined $g\left(x\right)$ is the inverse of the given function.

The graph of the inverse function is:

  

The graph of the function $f\circ g\left(x\right)$ is: