Implicit Functions

Every function met until now has been explicit: $y$ (or $x$) is isolated on one side, written directly as a formula in the other variable — $y=f(x)$ or $x=g(y)$. Many natural relationships between $x$ and $y$, however, cannot be rearranged into that form at all. If an equation links $x$ and $y$ together — for example $x^5+xy^3+x^2y+y^4=4$ — without either variable being solved for explicitly in terms of the other, then $y$ is said to be an implicit function of $x$ (and equally, $x$ is an implicit function of $y$). Such relations still define a curve, and $y$ can still vary smoothly with $x$ along it, so it sti …