Answer
Yes
Work Step by Step
Considering the given points in Exercise 98, find the slope of $f\left( x \right)$ as follows.
$\begin{align}
& m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \\
& =\frac{6.5-6}{7-6} \\
& =\frac{1}{2}
\end{align}$
Thus, the slope of $f\left( x \right)$is $\frac{1}{2}$.
Evaluate the function $f\left( x \right)$ as follows; it is enough to find that the line equation passes through the points $\left( 6,6 \right)$ with slope $\frac{1}{2}$.
$\begin{align}
& y-{{y}_{1}}=m\left( x-{{x}_{1}} \right) \\
& y-6=\frac{1}{2}\left( x-6 \right) \\
& y-6=\frac{1}{2}x-3 \\
& y=\frac{1}{2}x+3
\end{align}$
Therefore, the function $f\left( x \right)$ is $f\left( x \right)=\frac{1}{2}x+3$.
Considering the given points in exercise 98, find the slope of $g\left( x \right)$ as follows.
$\begin{align}
& m=\frac{{{y}_{2}}-{{y}_{1}}}{{{x}_{2}}-{{x}_{1}}} \\
& =\frac{8-6}{7-6} \\
& =\frac{2}{1} \\
& =2
\end{align}$
Thus, the slope of $g\left( x \right)$ is 2.
Evaluate the function $g\left( x \right)$ as follows; it is enough to find that the line of the equation passes through the points $\left( 6,6 \right)$ with slope $2$.
$\begin{align}
& y-{{y}_{1}}=m\left( x-{{x}_{1}} \right) \\
& y-6=2\left( x-6 \right) \\
& y-6=2x-12 \\
& y=2x-6
\end{align}$
Therefore, the function $g\left( x \right)$is $g\left( x \right)=2x-6$.
Evaluate $\left( f\circ g \right)\left( x \right)$ as follows.
$\left( f\circ g \right)\left( x \right)=f\left( g\left( x \right) \right)$
Use the function $g\left( x \right)=2x-6$,
$\left( f\circ g \right)\left( x \right)=f\left( 2x-6 \right)$
Use the function $f\left( x \right)=\frac{1}{2}x+3$,
$\begin{align}
& \left( f\circ g \right)\left( x \right)=\frac{1}{2}\left( 2x-6 \right)+3 \\
& =x-3+3 \\
& =x
\end{align}$
Evaluate $\left( g\circ f \right)\left( x \right)$ as follows.
$\left( g\circ f \right)\left( x \right)=g\left( f\left( x \right) \right)$
Use the function $f\left( x \right)=\frac{1}{2}x+3$,
$\left( g\circ f \right)\left( x \right)=g\left( \frac{1}{2}x+3 \right)$
Use the function $g\left( x \right)=2x-6$,
$\begin{align}
& \left( g\circ f \right)\left( x \right)=2\left( \frac{1}{2}x+3 \right)-6 \\
& =x+6-6 \\
& =x
\end{align}$
Therefore, $f\left( g\left( x \right) \right)=g\left( f\left( x \right) \right)=x$.
From the above mentioned definition of inverses of composite functions, it can be concluded that the functions f and g are inverses of each other.
