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

Question

Avery Alexander · Accepted Answer

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#

Jameson Allen · Answer

undefined To determine if the relation $ x^2 + y^2 = 25 $ defines a function, we can use the vertical line test. If any vertical line intersects the graph of the relation at more than one point, then the relation is not a function. If every vertical line intersects the graph at most once, then the relation is a function. Applying this test to the given relation, since every vertical line intersects the graph at most once (either at one point or not at all), the relation defines a function. Therefore, $ x^2 + y^2 = 25 $ defines a function.