Implicit functions

When a relation between $x$ and $y$ is given by means of an equation not solved for $y$, then $y$ is called an implicit function of $x$. For example, the equation

$$x^2 - 4y = 0$$

defines $y$ as an implicit function of $x$. Evidently $x$ is also defined by means of this equation as an implicit function of $y$. Similarly,

$$x^2 + y^2 + z^2 - a^2 = 0$$

defines anyone of the three variables as an implicit function of the other two.

It is sometimes possible to solve the equation defining an implicit function for one of the variables and thus change it into an explicit function. For instance, the above two implicit functions may be solved for $y$, giving $y = \frac{x^2}{4}$ and $y = \pm \sqrt{a^2 - x^2 - z^2}$; the first showing $y$ as an explicit function of $x$, and the second as an explicit function of $x$ and $z$. In a given case, however, such a solution may be either impossible or too complicated for convenient use.

The two implicit functions used in this section for illustration may be respectively denoted by $f(x,y) = 0$ and $F(x,y,z) = 0$.