Answer
The required values are $f\left( g\left( x \right) \right)=x$ and $g\left( f\left( x \right) \right)=x$. And the functions $f\left( x \right)=4x+9$ and $g\left( x \right)=\frac{x-9}{4}$ are inverses of each other.
Work Step by Step
Consider the functions:
$f\left( x \right)=4x+9$
and
$g\left( x \right)=\frac{x-9}{4}$
The equation for $f$ is given as:
$f\left( x \right)=4x+9$
Replace $x$ with $g\left( x \right)$
$\begin{align}
& f\left( g\left( x \right) \right)=4g\left( x \right)+9 \\
& =4\left( \frac{x-9}{4} \right)+9 \\
& =x-9+9 \\
& =x
\end{align}$
Now, to find $g\left( f\left( x \right) \right)$
Consider the function $g\left( x \right)$:
$g\left( x \right)=\frac{x-9}{4}$
Replace $x$ with $f\left( x \right)$
$\begin{align}
& g\left( f\left( x \right) \right)=\frac{f\left( x \right)-9}{4} \\
& =\frac{\left( 4x+9 \right)-9}{4} \\
& =\frac{4x}{4} \\
& =x
\end{align}$
Because $g$ is inverse of $f$ (and vice-versa), the inverse notation can be used:
$f\left( x \right)=4x+9$ and $\text{ }{{f}^{-1}}\left( x \right)=\frac{x-9}{4}$
It can be easily observed that ${{f}^{-1}}$ can be expressed in $f$ by subtracting $9$ and dividing by $4$ .
$\begin{align}
& \text{ }{{f}^{-1}}\left( x \right)=\frac{x-9}{4} \\
& =g\left( x \right)
\end{align}$
Hence, the required values are $f\left( g\left( x \right) \right)=x$ and $g\left( f\left( x \right) \right)=x$. And the functions $f\left( x \right)=4x+9$ and $g\left( x \right)=\frac{x-9}{4}$ are inverses of each other.
