How do you decide whether the relation x + y^3 = 64 defines a function?

Jan 4, 2016

It can be rewritten as $y = \sqrt[3]{64 - x}$, which uniquely determines $y$ for any Real value of $x$, so yes it is a function.

Explanation:

Given $x + {y}^{3} = 64$, subtract $x$ from both sides to get:

${y}^{3} = 64 - x$

Take cube roots of both sides:

$y = \sqrt[3]{64 - x}$

This uniquely determines $y$ for any Real value of $x$, so the relation does describe a function.

Also note that $x = 64 - {y}^{3}$ so it has a well defined inverse function too.

Footnote
$x + {y}^{3} = 64$ does not define a function if $x$ and $y$ are allowed to take Complex values, since every number has $3$ cube roots in the Complex plane, hence there would be three possible values of $y$ for each value of $x$.