I am taking Abstract Algebra and in the first assignment we were asked to determine which values of $n$ make the following function injective.
$f: \mathbb{R}\rightarrow \mathbb{R}$
$x \mapsto x^n$ $| n \in \mathbb{N^+}$
Obviously the case where $n$ is even is quite easy to disprove. For the odd case I had a more difficult time. I understand that I could use the fact that the odd functions are continuous and because their derivative is positive everywhere (besides 0 which is dealt with separately), the function is increasing so $a > b \implies f(a) > f(b)$ which would prove it was injective. However, I do not think this is the proper way to go about it, as we have not and will not cover continuity and those sort of things. Not to mention this method seems out of place with the other problems in the homework which all deal with equivalence relations and general set theory questions.
So I just want to know, is there a simpler way of going about proving the odd power function is injective that does not use much Real Analysis as much?