Question

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

Answer 1

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->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`