# How do you decide whether the relation x^2 + y^2 = 25 defines a function?

Nov 1, 2015

A relation is a function if for every $x$ there is (at most) one $y$.
A function can be seen as a recipe, saying if $x$ is such, then $y$ is so.

#### Explanation:

In this case the relation can be rewritten as
${y}^{2} = 25 - {x}^{2} \to y = + \sqrt{25 - {x}^{2}} \mathmr{and} y = - \sqrt{25 - {x}^{2}}$

These values are only defined in the domain $- 5 \le x \le 5$, but that's not important here:

For the $x$'s in the domain there are always TWO $y$'s (except when $x = - 5 \mathmr{and} x = 5$)

Extra:
This relation defines a circle with centre $\left(0 , 0\right)$ and radius $\sqrt{25} = 5$