Is the graph of a circle a function?

Answer 1

Yes, the graph of a circle is a function. Recall, in a circle, that x2 + y2 = r2, for circles at the origin, though you could offset it any place you want by making the equation more complex. Solving for y, you get y = square root(r2 - x2). Since square root, by definition, has two values, the plus value and the minus value, this is a function, even though each x has two values for y, within the domain of +/-R.

Some people might say that the graph of a circle is not a function, because it violates the single value test, i.e. if you draw a vertical line within the domain of the function, you get more than one value. In fact, however, if you convert the equation from cartesian coordinates to polar coordinates, you pass the single value test, i.e. R is constant for every theta.