Why can a circle be described by an equation but not by a function?

Solution 1:

Well, a circle can be described by a function, just not in the sense that you may be familiar with. If you are looking at a function that describes a set of points in Cartesian space by mapping each $x$-coordinate to a $y$-coordinate, then a circle cannot be described by a function because it fails what is known in High School as the vertical line test.

A function, by definition, has a unique output for every input. However, for almost all points on a circle, there is another point with the same $x$-coordinate. So, you would need your function to give two different $y$-coordinates for certain inputs, which is not allowed.

However, there is no rule that the input of a function has to be an $x$-coordinate or that the output has to be a $y$-coordinate, so we can define other functions that describle a circle. In more formal terms, the domain and codomain of a function do not have to be $\Bbb{R}$. For example, we can have a function that outputs an ordered pair (that is, codomain of $\Bbb{R}\times\Bbb{R}$). Then, $$f(t)=(\sin t,\cos t)$$ outputs the unit circle when $0\le t<2\pi$. We could also describe the points in space in a different way, using polar coordinates. Here we use the counter-clockwise angle from the positive $x$-axis, $\theta$, and the distance from the origin, $r$, to identify a point. Using this system, we can easily describe the unit circle as $(\theta,f(\theta))$, where $f(\theta)=1$ and $0\le\theta<2\pi$.

Solution 2:

You say that "functions look like glorified equations". There's definitely some truth to that. Here's an equation:

$$y = x^2$$

Here's another equation:

$$f(x) = x^2$$

Both of these are equations, and the two equations do very similar things. The two equations both define functions. But they do it a little bit differently.

The first equation doesn't give a name to the function it defines; it just defines $y$ as being "a function of" $x$, which is to say that it tells you what $y$ is once you know what $x$ is. The second equation does give a name to the function it defines; the function is called $f$. (The function isn't $f(x)$; the function is just $f$.)

Now, here's another equation:

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

Unlike $y = x^2$, this equation does not define a function. Why not? Because it doesn't tell you what $y$ is once you know what $x$ is. This equation defines a relation, which is to say that it tells you what the values of $x$ and $y$ are allowed to be, but the value of $y$ is not completely determined by the value of $x$.

A function, it turns out, is just a special kind of relation. A function is any relation that has the property that once you know what the first value is, you know what the second value is. A circle can be described by a relation (which is what we just did: $x^2 + y^2 = 1$ is an equation which describes a relation which in turn describes a circle), but this relation is not a function, because the $y$ value is not completely determined by the $x$ value.

Now, could we use something similar to function notation in order to define a circle? Sure. What we can't do is something like this:

$$x^2 + f(x)^2 = 1$$

Since we're using function notation here, it looks like we're still trying to define a function. But what we can do is give our relation a name. Let's call it $\diamond$. Now we can say this:

$$x \diamond y \text{ whenever } x^2 + y^2 = 1$$

Now, much like $f$ is the name of a function defining a parabola way above, $\diamond$ is the name of a relation defining a circle.