Question

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

Answer 1

`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.

Explanation:

Let's write this equation in a different form.

`y^2-(1-x^2)=0`.

Now think of it like the difference of two squares and write this as the product of two binomials.

`(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.