A multi-additive function!

By intuition, one can infer that the function is actually an exponential function, and with more intuition, one can infer that this was written for the problem writing party. The function I intended to use was $f(x) = 2^x$ . This comes from the fact that $2^{x+y} = 2^x \cdot 2^y$

The more formal solution:

$f(3) = f(2+1) = f(2) \cdot f(1) = f(1+1) \cdot f(1) = f(1)\cdot f(1)\cdot f(1) = f(1)^3 = 8$

This gives us $f(1) = \sqrt[3]{8} = 2$

Also, $f(11) = f(1+1+1+...+1)$ with 11 $1$ 's, giving us $f(11) = f(1)^{11} = 2^{11} = 2048$

Then, $f(5) = f(1)^5 = 2^5 = 32$

So, $f(11)-f(5) = 2048-32 = \boxed{2016}$