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

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.

${y}^{2} - \left(1 - {x}^{2}\right) = 0$.

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

$\left(y - \sqrt{1 - {x}^{2}}\right) \left(y + \sqrt{1 - {x}^{2}}\right) = 0$

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.