a circle graph is not a function?

The definition of a function is so important. In addition to the above, the picture below (taken from: What is a function) may help.

(the left hand side is your X and the right hand side is the value Y)

A function is a rule that assigns uniquely to a member of domain set, a member of the image set. The key word is "uniquely". So if you assign say 2 as well as -2 to number 1, then you have a rule, but not a function. That is the logic behind the vertical line test. If you draw a vertical line and it intersects the graph of the function in two distinct points, then you can see that it means I have assigned both of these points to the point where my vertical line crosses the x-axis. An example of this is the circle.

However a semi-circle is a legit function-the upper half is the positive square root (y=+$\sqrt{1-x^2}$) and the bottom half is the negative square root (y=-$\sqrt{1-x^2}$).

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 ($y=\pm\sqrt{\dfrac34}$).