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

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Answer 1 · 2015-11-01T23:20:54

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.

In this case the relation can be rewritten as
$y^2=25-x^2->y=+sqrt(25-x^2)ory=-sqrt(25-x^2)$

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

For the $x$ 's in the domain there are always TWO $y$ 's (except when $x=-5orx=5$ )

Extra:
This relation defines a circle with centre $(0,0)$ and radius $sqrt25=5$

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