Finding values for $x$ and $y$ given ONE equation

Question

Ok so I'm in precalculus right now and the directions on my homework seem to make no sense to me.

I'm asked to find values for $x$ and $y$ given this equation: $y=x^{1/3}$

Doesn't this mean I could have infinite values for $x$ and $y$ because I only have one equation? I'm not sure what my professor is looking for.

NOT looking for an answer, simply an explanation.

Answer 1

Yes, you can infinitely many ordered pairs $(x,y)$. For example, one such pair would be $(8,2)$.

Answer 2

You are correct that there is not a pre-determined, finite collection of $(x,y)$ pairs that satisfy this equation. In that sense, the instruction to "find" pairs is misleading. What is almost certainly intended is that you select, as you please, some values of either $x$ or $y$, and then determine, from the equation, what the other value of the pair must be.

You can take the cube root of any real $x$, positive or negative, and similarly, you can cube any real $y$, so there are, in this case, no limits on the value you can pick.

Frequently the next step is to consider the $(x,y)$ pairs as Cartesian co-ordinates of points, and plot all these points on a graph

Answer 3

You can take any perfect cube number as the value of $x$ and it's cube root as the value of $y$.

For example: $(x,y) = (3,1)$ ; $(8,2)$ ; $(27,3)$ ; any $(x, \sqrt{x})$ could be the answer.