Why can a circle be described by an equation but not by a function? In high-school math functions always looked to me just like glorified equations. The only time I saw a meaningful difference was when we covered the equation of a circle and I realized that an equation can describe a circle but a function cannot (a function can describe only one half of the circle).
So what is the difference between a function and an equation that leads to this, in formal terms?
 A: This is sort of an artifact of the way we draw functions. We place (input, output) pairs on the plane through the arbitrary process 


*

*Start at the center of the paper.

*Travel [input] distance to the right

*Travel [output] distance up

*Place a dot on the paper.

*Repeat for all possible inputs


You can see that this process can never draw a circle because at step 3, we have to go either up or down depending on whether [output] is positive or negative. But for a circle we would have to do both!  
Imagine instead we did this instead:


*

*Imagine a horizontal line through the center of the paper

*Imagine a ray with vertex in the center of the paper, making an angle of [input] from the horizontal line

*Travel [output] distance along the ray

*Place a dot on the paper  

*Repeat for all possible inputs


Then a circle of radius $r$ would very easily be represented by the function $f(x) = r$. (This is commonly known as polar coordinates)
Lastly, the way we usually draw equations is this:


*

*Pick a point on the paper

*If the point's horizontal and vertical distance to the center of the paper satisfy the equation, draw a dot on the paper.

*Repeat for all points on the paper  


You can see that this method does not have the same restriction as the first method, and therefore the image of a circle could result. 
A: When the relation is univocal (one $x$ gives one $y$), you can use a function, like $y=f(x)$.
When the relation is not univocal (one $x$ gives several $y$), you need several functions, like $y_0=f_0(x),y_1=f_1(x)$.
This can be combined in a single equation like $(y-f_0(x))(y-f_1(x))=0$.
In the case of a unit circle, for example, $$y_0=\sqrt{1-x^2},\\y_1=-\sqrt{1-x^2},$$ or $$(y-\sqrt{1-x^2})(y+\sqrt{1-x^2})=0,$$ i.e. $$x^2+y^2=1.$$
More generally, an implicit equation like $F(x,y)=0$ gives several $y$ for one $x$ and several $x$ for one $y$.
A: 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 $0\le\theta<2\pi$.
A: 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.
A: If you allow functions to misbehave a bit (multi-valued functions) like square roots for example, then a circle can be described as a function. (e.g. the square root of 9 is both +3 and -3 (since both values, when squared, yield 9)). 
E.g. solving $x^2 + y^2 = r^2$ for y:
$y = \sqrt{r^2 - x^2}$
Outside the circle (i.e. when x < -r or x > r), y is undefined, but within the circle, the square root above has two roots forming the top and bottom parts of the circle. 
Hmmm - I don't have enough rep to comment, hence the answer.
