explain why the equation x^2 + y^2=1
does not define y as a function of x.
Well, if you solve...
y= +- sqrt (1-x^2)
which means for any value of x, there are two values (the + and the -) of y. To be a function, there has to be one to one correspondence, that is, for any x, there can be only one y value if y is a function of x.