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Piecewise-Defined Function

A piecewise-defined function is one which is defined not by a single equation, but by two or more. Each equation is valid for some interval .

Example 1:

Consider the function defined as follows.

$y = {\begin{cases} x + 2 for x < 0 \\ 2 for 0 \leq x \leq 1 \\ - x + 3 for x > 1 \end{cases}$

Math diagram

The function in this example is piecewise-linear, because each of the three parts of the graph is a line.

Piecewise-defined functions can also have discontinuities ("jumps"). The function in the example below has discontinuities at $x = - 2$ and $x = 2$ .

Example 2:

Graph the function defined as shown.

$y = {\begin{cases} \frac{1}{2} x^{2} for x < - 2 \\ 0for - 2 \leq x < 2 \\ \frac{1}{2} x^{2} for x \geq 2 \end{cases}$

Note that we use small white circles in the graph to indicate that the endpoint of a curve is not included in the graph, and solid dots to indicate endpoints that are included.

Math diagram

Example 3:

Graph the function defined below.

$y = {\begin{cases} \log x for 0 < x < 1 \\ \frac{1}{x - 2} for x \geq 1 \end{cases}$

Math diagram

Negative values of $x$ and $0$ are not included in the domain because the first function, $\log x$ , is undefined for those values. The value $x = 2$ is not included in the domain because the second function is not defined for that value (it has a vertical asymptote there). Therefore the domain of this function is ${x | 0 < x < 2} \cup {x | x > 2}$ . This can be represented using interval notation as $(0, 2) \cup (2, \infty)$ .