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# Implicit functions

A curve can be defined implicitly by a relation of the form $$F\big(y(x),x\big) = 0.$$

For instance, a general function $y=f(x)$ can be rewritten as $y-f(x) = 0$.

A circle is defined implicitly by the relation $$y^2 + x^2 = 1.$$

The curve is not defined directly as a function of $x$. It must be deduced from the equation, and in general corresponds to the graph of several explicit functions.

This method of defining a curve is used for those where it would be too complex to write the equation explicitly.

For instance, the circle combines the graphs of two explicit functions $$y_+=\sqrt{1-x^2},\quad y_-=-\sqrt{1-x^2},\qquad x\in[-1,1].$$

Implicit and explicit equation of a circle