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

Answer 1

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.

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

#y^3=64-x#

Take the two sides' cube roots:

#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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Answer 2

Eva Abramson

To determine whether the relation (x + y^3 = 64) defines a function:

Solve the equation for (y) to see if there is a unique (y) value for each (x).
If there is only one (y) value for each (x), then the relation defines a function. If there are multiple (y) values for any (x), then the relation does not define a function.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.