Solution

Algebraically, this question is tough. The goal (algebraically) for this type of "is ___ a function of __" is to rewrite the equation so that the first variable is solved for. In this case, that would mean solving for \(y\). However, if you have already tried this, you should notice that no matter what algebraic trick you attempt, isolating \(y\) as a variable just doens't seem to work. Here are some "failed" attempts:

\[ \begin{{array}}{{rcl}} x^2y^3-y &=& 1\\ y(x^2y^2-1) &=& 1 \\y &=&\displaystyle\frac{{1}}{x^2{\color{{red}} y}^2-1} \\ \hline x^2y^3-y &=& 1\\ x^2y^3 &=& 1+y \\ x^2 &=& \displaystyle\frac{1+y}{y^3}\end{{array}} \]

This is by no means exhaustive of all the possible tricks we might employ, but certainly once you have attempted a few different algebraic approaches, choosing a visual approach instead makes plenty of sense. In this case, we can graph the function and seek to identify if the fails the vertical line test anywhere. Given our algebraic difficulties, it is reasonable to assume that might be the case.

Our first graph doesn't make this super clear.

However, zooming in a bit we can see that this most definitely fails the vertical line test, therefore \(y\) is not a function in \(x\) for the given equation.