What is the inverse function of 0.5^2?
As emakarov said, in order to have an inverse function, you have to have a function! $\displaystyle 0.5^2= 0.25$ is a number, not a function. If you mean $\displaystyle f(x)= x^2$, for x non-negative, then its inverse is $\displaystyle f^{-1}(x)= \sqrt{x}$. And, of course, $\displaystyle f^{-1}(0.25)= 0.5$. As long as f has an inverse, if $\displaystyle f(a)= b$, then $\displaystyle f^{-1}(b)= a$.
I tried it and it looks right to me... Remember that the graph of the inverse function will look like it's been reflected along the x=y line. This doesn't mean there won't be any overlap of the functions, and it doesn't mean the functions will be in opposing quadrants, which I think is what you're looking for.
Think about it. If f(x) returns y, then g(y) returns x if g(x) and f(x) are inverse functions. In this case, where $\displaystyle f(x) = y = 2^{(-x)}$, you can have any real x you want, but y will always be positive, no matter what your x value is. So basically, the function f(x) in this case takes ANY x value, and returns a positive y value. What does that mean about its inverse function? Well... that means that its inverse function will take any positive y value, and return some real x value (no restrictions). Since you're taking the inverse function of x, that means in this case your x will always be positive and your y will span across the real numbers.
Look at your graph again. Does it look like a reflection about the line y=x?