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