I know to be surjective , every real number in range maps to some real number in the domain of this function. Since isolating x in terms of y seems a bit tricky, I was thinking of using proof cases for specific values of y where the slope increases and decreases. I'm not sure, assuming this is the proper technique for this proof, how to rearrange x in terms of y?
edit: cannot use external facts/theories
You are far better off using the intermediate value theorem than trying to find a preimage for each real number. In general, this usually wont even be possible.
Let $r$ be a real number. Since $\lim\limits_{x\to \infty}(x^5-x)=\infty$ and $\lim\limits_{x\to -\infty}(x^5-x)=-\infty$, there exists $N$ such that $g(-N)<-r$ and $g(N)>r$. Now $g$ is continuous on the interval $[-N,N]$, and since $r\in [g(-N),g(N)]$, there exists $s\in [-N,N]$ such that $g(s)=r$.