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

Question

Hazel Anglin · Accepted Answer

#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.

Here's another way to write this equation. #y^2-(1-x^2)=0#. Now write this as the product of two binomials and think of it as the difference of two squares. #(y-sqrt(1-x^2))(y+sqrt(1-x^2))=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.

Jameson Allen · Answer

undefined To determine whether the relation $x^2 + y^2 = 1$ defines a function, we use the vertical line test. If every vertical line intersects the graph of the relation at most once, then the relation is a function. If any vertical line intersects the graph at more than one point, the relation is not a function. Applying the vertical line test to the equation $x^2 + y^2 = 1$, we find that every vertical line intersects the graph at most once, indicating that the relation defines a function. Therefore, $x^2 + y^2 = 1$ represents a function.