Answer
a) ${{f}^{-1}}\left( x \right)=x-5$
b) they are inverses
Work Step by Step
(a)
Consider the function, $f\left( x \right)=x+5$
Step-1-Replace $f\left( x \right)$ with $y$.
$\begin{align}
& f\left( x \right)=x+5 \\
& y=x+5
\end{align}$
Step-2-Interchange $x$ and $y$
$\begin{align}
& y=x+5 \\
& x=y+5 \\
\end{align}$
Step-3-Solve for $y$. Subtract $5$ from each side
$\begin{align}
& x=y+5 \\
& x-5=y+5-5 \\
& x-5=y
\end{align}$
Step-4-Replace $y$ in step 3 by ${{f}^{-1}}\left( x \right)$
$\begin{align}
& y=x-5 \\
& {{f}^{-1}}\left( x \right)=x-5
\end{align}$
Therefore, the required inverse of the function is ${{f}^{-1}}\left( x \right)=x-5$
(b)
Consider the functions:
$f\left( x \right)=x+5$ and ${{f}^{-1}}\left( x \right)=x-5$
Replace $x$ with ${{f}^{-1}}\left( x \right)$ in \[f\left( x \right)\]
$\begin{align}
& f\left( {{f}^{-1}}\left( x \right) \right)={{f}^{-1}}\left( x \right)+5 \\
& =\left( x-5 \right)+5
\end{align}$
Simplify,
$\begin{align}
& f\left( g\left( x \right) \right)=\left( x-5 \right)+5 \\
& =x
\end{align}$
Now, find ${{f}^{-1}}\left( f\left( x \right) \right)$
The equation for ${{f}^{-1}}\left( x \right)$ is given as:
${{f}^{-1}}\left( x \right)=x-5$
Replace $x$ with $f\left( x \right)$
$\begin{align}
& {{f}^{-1}}\left( f\left( x \right) \right)=f\left( x \right)-5 \\
& =\left( x+5 \right)-5
\end{align}$
Simplify
$\begin{align}
& {{f}^{-1}}\left( f\left( x \right) \right)=\left( x+5 \right)-5 \\
& =x
\end{align}$
Thus, $f\left( {{f}^{-1}}\left( x \right) \right)=x$ and ${{f}^{-1}}\left( f\left( x \right) \right)=x$
It can be easily observed that ${{f}^{-1}}$ can be expressed in $f$ if subtracted by 5.
Hence, the functions $f$ and \[{{f}^{-1}}\] are inverses of each other.
