Power function and involution $f(x) = x^a$

Question

For power functions we have a variable $x$ and a constant $a$; we get that $f(x) = x^a$. Find all involutions for $f(x)$.

I started out with basic functions such as $f_1(x) = f_1^{-1}(x) = x^1$ and $f_2(x) = f_2^{-1}(x) = x^{-1}$. Is that all of them? How could I be sure of that?

Edit 1.

Problem description: Is there any involutions for power functions? (i.e. $x^a)$ If there is, find all of them.

Answer 1

You're definitely on the right track. Here's a hint: you want to show that $f(f(x)) = x$; equivalently,

$$x = f((x^a)) = (x^a)^a = x^{a^2}.$$

Can you take it from here?