a circle graph is not a function? I'm a little confused by the rule: If you draw a vertical line that intersects the graph at more than 1 point then it is not a function.
Because then a circle like $y^2 + x^2 = 1$ is not a function? 
And indeed if I rewrite it as $f(x) = \sqrt(1 - x^2)$ then wolfram alpha doesn't draw a circle. I guess I'm missing the intuition as to why this is though?

Functions need to be well-defined as part of their definition, so for a given input there can only be one output.
$f(x,y)=x^2+y^2-1$ is a function of two variables, and the set of points for which this function gets $0$ is the unit circle.
However writing $y^2+x^2=1$ as a function of $x$ alone cannot be done, as $x=\dfrac12$ has two solutions ($y=\pm\sqrt{\dfrac34}$).
