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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
Implicit and explicit equation of a circle