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Piecewise-Defined Function

Apiecewise-definedfunction is one which is defined not by a single equation, but by two or more. Each equation is valid for someinterval.

Example 1:

Consider the function defined as follows.

y = { x + 2 for x < 0 2 for 0 ≤ x ≤ 1 − x + 3 for x > 1

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 = { 1 2 x 2 for x < − 2 0for − 2 ≤ x < 2 1 2 x 2 for x ≥ 2

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.

Example 3:

Graph the function defined below.

y = { log x for 0 < x < 1 1 x − 2 for x ≥ 1

Negative values of x and 0 are not included in thedomainbecause the first function, log x ,是未定义的ose 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 } ∪ { x | x > 2 } . This can be represented using interval notation as ( 0 , 2 ) ∪ ( 2 , ∞ ) .