Just wondering why we have equations defined in the Cartesian coordinate for circles and ellipses: wouldn't graphing those shapes contradict the fundamental property of a function (i.e.: to each element in the domain corresponds a single element in the codomain)?
I know I can algebraically restate the equations as a function of y, but that would leave me with a square root whose domain I don't know, so the problem remains.
On
The equation of a conic describes a set of points in the Cartesian plane, $\{(x,y)|f(x,y)=0\}$, where $f$ is a function of two variables. You're right that for many conics there's no function $g:x\mapsto y$ of one variable that describes the same set of points. That doesn't diminish the utility of those equations, though.
Conic sections are relations, not functions.
An equation (like for example $x^2 + y^2 = 100$) can be thought of as a criterion or condition that can be used to check any point to see if it is part of the relation. For this example, $(7,4)$ is not part of the relation (because $7^2 + 4^2 ≠ 100$) but $(6, 8)$ is part of the relation (because $6^2 + 8^2 = 100$).
A function is a relation with the special, additional property that to each $x$ value in the domain there is one and only one corresponding $y$ value. But relations in general don't have to meet that additional requirement; in a general relation, there can be lots of different $y$ values corresponding to a single $x$ value.