Proof of property of given homomorphism

Question

Given that $f$ to be group homomorphism from $\mathbb R^*\to\mathbb R^*$ and I want proof of property "If $x>0$ then $f(x)>0$". Please give me hint or something so I can proceed further.

Answer 1

Hint: For $a \in \Bbb R^*$, we have $f(a^2) = f(a)^2$

Answer 2

If $x>0$, then $$x=\sqrt{x}\sqrt{x}$$ so $$f(x)=f(\sqrt{x}.\sqrt{x})=f(\sqrt{x})f(\sqrt{x})=\Big(f(\sqrt{x})\Big)^2 >0$$

Additional exercise: Prove in general that every automorphism of $\Bbb{R}^*$ maps positive numbers to positive numbers and negative numbers to negative numbers