How do you decide whether the relation x^2 + y^2 = 1 defines a function?

1 Answer
Mr. Mike
Apr 23, 2018

x^2+y^2=1 does not describe a function because there exist valid values of x for which more than one value of y make the equation true.

Explanation:

Let's write this equation in a different form.

y^2-(1-x^2)=0.

Now think of it like the difference of two squares and write this as the product of two binomials.

(y-sqrt(1-x^2))(y+sqrt(1-x^2))=0

Note that there are TWO solutions for y here, namely

y=sqrt(1-x^2), and y=-sqrt(1-x^2).

This relation is NOT a function. In order for an equation to represent a function, every x in the range of the function must only have one y-value.

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