Answer
$f^{-1}(x)=\displaystyle \frac{1}{4}x-\frac{9}{4}$
(the inverse is blue on the graph)
Work Step by Step
Step 1: Replace $f(x)$ with $y$.
$y=4x+9$
Step 2: Interchange $x$ and $y$.
$x=4y+9$
Step 3: Solve the equation for $y$.
$ x=4y+9,\qquad$ ... add $-9$,
$ x-9=4y,\qquad$ ... divide with $4$
$\displaystyle \frac{1}{4}x-\frac{9}{4}=y$
Step 4: Replace y with the notation $f^{-1}(x)$.
$f^{-1}(x)=\displaystyle \frac{1}{4}x-\frac{9}{4}$
Graphing $f(x)=4x+9, \left[\begin{array}{lll}
x & f(x) & (x,y)\\
0 & 9 & (0,9)\\
-3 & -3 & (-3,-3)
\end{array}\right], $
the graph of $f(x)$ is a line passing through $(0,9)$ and $(-3,-3)$.
The graph of $f^{-1}(x)$ is a line passing through points (y,x) of the above table,
$(9,0)$ and $(-3,-3)$.
