How to prove $\sup(f(x)^2) = (\sup(f(x)))^2$

Question

I started the proof with this:

$$\sup(f(x)^2) = \sup(f(x)*f(x)) = \sup(f(x)) * \sup(f(x) = (\sup(f(x)))^2$$

But it doesn't feel right. Any ideas or comments?

Answer 1

Let $f:[-10,1] \to \mathbb{R}$ such that $f(x)=x$. Then $sup(f(x)^2)=100$ and $(sup(f(x)))^2=1$. They are not equal.