Given that $f(3)=5$ and $f(3x)=f(x)+2$ for all $x$, find $f^{-1}(11)$.

If  f(3)  = 5   and    f(3x)  = f(x) + 2, then .....

f(3x)  = f(3) + 2     implies that x  = 3

So

f(3 * 3)  = f(3) + 2

f(9)  =  5 + 2

f(9)  = 7

And

f(3x)  = f(9) + 2   implies that  x  = 9...so.....

f(3*9) = f(9) + 2

f(27)  = 9

And

f(3x)  = f(27) + 2   implies that x  = 27

f(3 * 27)  = f(27) + 2

f(81)  = 9 + 2

f(81) = 11

And   (81, 11)  is on the original graph  .....so  (11, 81)  is on the inverse

So

f^-1(11)  =  81