Answer
See the explanation below.
Work Step by Step
(a)
Now, consider $f\left( x \right)=4x-3$.
Step 1: Replace $f\left( x \right)$ with $y$:
$y=4x-3$
Step 2: Interchange $x$ and $y$:
$x=4y-3$
Step 3: Now solve for the value of $y$:
$x+3=4y$
That is,
$y=\frac{x+3}{4}$
Step 4: Replace $y$ with ${{f}^{-1}}\left( x \right)$:
${{f}^{-1}}\left( x \right)=\frac{x+3}{4}$
Therefore, the inverse function ${{f}^{-1}}\left( x \right)$ of the function $f\left( x \right)=4x-3$ is ${{f}^{-1}}\left( x \right)=\frac{x+3}{4}$.
(b)
Consider the function, $f\left( {{f}^{-1}}\left( x \right) \right)$
$\begin{align}
& f\left( {{f}^{-1}}\left( x \right) \right)=f\left( \frac{x+3}{4} \right) \\
& =4\left( \frac{x+3}{4} \right)-3 \\
& =x+3-3 \\
& =x
\end{align}$
Next consider the function, ${{f}^{-1}}\left( f\left( x \right) \right)$
$\begin{align}
& {{f}^{-1}}\left( f\left( x \right) \right)={{f}^{-1}}\left( 4x-3 \right) \\
& =\frac{4x-3+3}{4} \\
& =x
\end{align}$
Hence, $f\left( {{f}^{-1}}\left( x \right) \right)=x$ and ${{f}^{-1}}\left( f\left( x \right) \right)=x$ for the function $f\left( x \right)=4x-3$.
