Piecewise Defined Functions

A piecewise-defined function is specified by different explicit formulas on different sub-intervals of its domain. Piecewise functions are crucial for analyzing limits, continuity, and step discontinuities.

Quick Reference

Function Type	Formula Structure	Example
Piecewise Function	Different formulas on sub-intervals	f(x) = \begin{cases} f_1(x) & x \in I_1 \\ f_2(x) & x \in I_2 \end{cases}
Absolute Value	Piecewise linear definition	\|x\| = \begin{cases} x & x \ge 0 \\ -x & x < 0 \end{cases}

Definition and Examples

A function defined by different formulas on different parts of its domain is called a piecewise-defined function.

A function f is piecewise-defined if its domain is partitioned into two or more disjoint sub-intervals, and on each sub-interval f is given by a distinct expression.

A classic example of a piecewise-defined function is the absolute value function:

|x| = \begin{cases} x & \text{if } x \ge 0 \\ -x & \text{if } x < 0 \end{cases}

Consider the piecewise-defined function:

f(x) = \begin{cases} 1 - x & \text{if } x \le 1 \\ x^2 & \text{if } x > 1 \end{cases}

Evaluate $f (0)$ , $f (1)$ , and $f (2)$ .

Solution

For $x = 0 \leq 1$ , use the first formula: $f (0) = 1 - 0 = 1$ .
For $x = 1 \leq 1$ , use the first formula: $f (1) = 1 - 1 = 0$ .
For $x = 2 > 1$ , use the second formula: $f (2) = 2^{2} = 4$ .

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Definition and Examples

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