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

1 Answer
George C.
Jan 4, 2016

It can be rewritten as y = root(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 = root(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.

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