Must a function link every input value to a single output value?

Question

I'm wondering whether equations like $x^2 + y^2 = 4$ can describe a function or not. The reason is that a function should normally link every input value to a single output value. However, in case of $x^2 + y^2 = 4$, when $x=0, y=2$ and $y=-2$. Likewise, when $y=0, x=2$ and $x=-2$, etc. So does the equation $x^2 + y^2 = 4$ describe a function? Thanks a lot in advance!

Answer 1

$x^2 + y^2 = 4$ is an equation, not a function! Often, people seem to use functions and equations interchangeably, but this is a shorthand. They're not the same thing. This is actually something that tripped me up a lot in high school.

For starters, yes, a function must map each input on its domain to exactly one output. No more, no less. $f(x)$ must always have an unambiguous value, when $x$ is in the domain of $f$. If $f$ maps an input to more than one value, we usually call it a relation, not a function. And if some inputs from the domain don't get mapped to anything, we call it a partial function.

But there's another issue here, about whether equations can be used interchangeably with functions. Someimes, it seems like they can. For instance, people sometimes say something like "Let $f(x)$ be the function $y = 4x + 2$". What they actually mean is, "Let f be the function that takes a number $x$, and produces $4x + 2$". It seems like a silly difference, but it becomes important in more complicated examples like the one you asked about.

Another way of thinking about it is that the equation puts a constraint on the function in question. So, we declare the existence of some function $f$, from real numbers to real numbers, but we don't know anything else about it yet. Then we declare that $f$ satisfies this particular constraint: that if $x$ is a real number, then $f(x) = 4x + 2$. And there is exactly one function satisfying that constraint.

What happens when we try that with $x^2+f(x)^2=4$? Well, that constrains $f$ to be a function on the interval [-2, 2], if we're using real numbers still. And any point (x, y) on the circle of radius 2 around the origin will satisfy this equation. But it doesn't sufficiently constrain the function!

We don't know, for example, whether $f(0)$ should be 2 or -2, and we have to pick one, or else $f$ won't be a function. So, sometimes an equation is enough to precisely describe a function, and sometimes it isn't. That's ok, because equations are not functions.

One last note: @vgg above mentioned the square root function, $\sqrt{x}$. In order to make this a function, it is usually defined to be the positive root of $x$. So $-2 \neq \sqrt{4}$.