Prove or disprove: $f(x)=\ln(x^2)$ is surjective or injective

Question

Consider the function $f:(0,\infty)→(−\infty,\infty)$ defined by $f(x)=\ln(x^2)$. Prove or disprove each of the following statements.

1. The function $f$ is surjective.
2. The function $f$ is injective.

Any help would be nice

Answer 1

Hint: show the following much more interesting fact:

For a composition of functions $f\circ g$ it holds:

• if $f\circ g$ is injective, then $f$ is.
• if $f\circ g$ is surjective, then $g$ is.

Both can be easily proved by contradiction.