what considers a valid function

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I remember been told back in high school that for a function to be valid, one input has to correspond with one output and vice versa.

But nowadays, I encounter function such as the function of circle,

$$x^2 + y^2 = 1$$

which clearly does not conform to the one input one output rule.

So I was wondering, what is considered a valid function?

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1 Answer

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For something to be a function, you need one output for each input. You do not need vice versa. Your equation $x^2+y^2=1$ does not define $x$ or $y$ as a function of the other because (for example) if I give you an $x$ there are $0$ or $2$ values for $y$ that satisfy the equation. If you write $y=\sqrt{1-x^2}$ for $-1 \le x \le 1$ you have a valid function because we have restricted the $y$ values to $[0,1]$ and for any $x$ in the domain there is only one $y$ in the range that corresponds. It is still not vice versa because for $y=\frac 35$ you can have either $x=\frac 45$ or $x=-\frac 45$, but that is not a problem.

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