Question

Is the equation `x^2+y^2=4` a function?

Answer 1

No - there is more than one `y` value for each `x` value in the domain.

Explanation:

Equations in the form of `ax^2+by^2=r^2` form circles.

`x^2+y^2=4=2^2` looks like this:

graph{x^2+y^2-4=0 [-7.023, 7.024, -3.51, 3.513]}

Is this a function? That is, does there exist a single `y` value for each `x` value in the domain? And the answer is no - for essentially all the `x` values (excepting `x=pm2`), there are two `y` values for each `x` value.

This is often called the "vertical line test" - if you draw a vertical line for any value of `x` in the domain, does the vertical line encounter no more than one `y` value. We can see that if we do that (such as draw a line along the `y` axis), we'll hit two `y` values.