How do I solve $x^5 +x^3+x = y$ for $x$? I understand how to solve quadratics, but I do not know how to approach this question. Could anyone show me a step by step solution expression $x$ in terms of $y$?
The explicit question out of the book is to find $f^{-1}(3)$ for $f(x) = x^5 +x^3+x$
So far I have reduced $x^5 +x^3+x = y$ to $y/x - 3/4 = (x^2 + 1/2)^2$ or $y = x((x^2+1/2)^2 + 3/4)$ but Im still just as lost.

A few of the answers suggest guessing the answer by inspection. Indeed, in most calculus problems that could be on an exam it is very likely that the solution to the polynomial is meant to be 'obvious', frequently one of $0,\pm1,\pm2$. This is just to keep numbers and computations reasonable, but is not an absolute guarantee (especially if you get to use a calculator). The mantra is usually "the uglier it looks, the more likely it is that there is a simple answer and argument."
But there's also a formal justification. Our first hope would be for a rational solution to $f(x)=3$. Sounds like a lot of possibilities, right? But by the rational roots theorem the only possibilities here are $\pm1,\pm3$. That's a nice, small list, and it's easy to verify that one of these actually works, thus solving the problem. 
If there wasn't a rational root we'd be stuck with a more problematic situation, but it's always a good idea to check for predictable, nice answers first, especially when it involves so little effort to do so.
