The function has no inverse, because it is not one-to-one. Proving this is not so easy. To prove it directly, we would have to come up with an x and y, not equal to each other, such that:
x^x = y^y
We can prove it slightly less directly by noting that:
f(0.2) = 0.72477966367769553147061400088157
f(0.5) = 0.70710678118654752440084436210485
f(1) = 1
Now, f(x) = x^x is a continuous function (this can be proven by looking it as e^(x * ln(x))). By the intermediate value theorem, there must be some c inbetween 0.2 and 0.5 such that f(c) = 0.71, and some d between 0.5 and 1 such that f(d) = 0.71. However, c and d lie in different regions, so c does not equal d. Therefore, we have distinct c and d such that f(c) = f(d), proving that f(x) is not one-to-one, and hence does not have an inverse.