Home page for accesible maths 3 Chapter 3 contents 3.19 Basic examples of the Chain Rule 3.21 Contours

Style control - access keys in brackets

Font (2 3) - + Letter spacing (4 5) - + Word spacing (6 7) - + Line spacing (8 9) - +

3.20 Implicitly defined functions

$\bullet$ graphs $\longleftrightarrow$ functions

$\bullet$ curves $\longleftrightarrow$ implicit functions

The graph of a function has the special property that each value of $x$ is associated with only one value of $y$ ; whereas for a typical curve, some values of $x$ correspond to many values of $y$ . Curves in the plane correspond to implicitly defined functions, just as graphs correspond to functions. We can obtain curves by putting together graphs.

Example (Circle).

The formula $x^{2}+y^{2}-1=0$ defines a circle that we can regard as the union of the semicircles $y_{2}=\sqrt{1-x^{2}}$ and $y_{1}=-\sqrt{1-x^{2}}$ , where $y_{2}$ and $y_{1}$ are functions of $x\in[-1,1]$ .