Step 1 :Let's replace $f(x)$ with $y$ for simplicity, so we have $y=\frac{1}{9}x+2$
Step 2 :In order to invert $f(x)$ we may solve this equation for $x$. That gives $y-2=\frac{1}{9}x$
Step 3 :Multiplying both sides by 9 gives $9(y-2)=x$
Step 4 :Writing this in terms of $x$ gives the inverse function of $f$ as $f^{-1}(x)=\boxed{9(x-2)}$