Answer
$f(x)=\dfrac{x-1}{3}$
Work Step by Step
Let's note the inverse function:
$$f^{-1}(x)=3x+b.$$
Find the inverse $f$ of the function $f^{-1}$:
$$\begin{align*}
f^{-1}(x)&=3x+b\quad&&\text{Write original function.}\\
y&=3x+b\quad&&\text{Replace }f^{-1}(x)\text{ by }y\\
x&=3y+b\quad&&\text{Switch }x\text{ and }y.\\
x-b&=3y\quad&&\text{Subtract }b\text{ from each side. }\\
\dfrac{x-b}{3}&=y\quad&&\text{Divide each side by }3.
\end{align*}$$
The function $f$ is $f(x)=\dfrac{x-b}{3}$.
For example, for $b=1$, $f(x)=\dfrac{x-1}{3}$.
