Lesson 2: Piecewise-Defined Functions

Property

A piecewise-defined function is a function that uses different rules (expressions) for different intervals of its domain. The standard notation uses cases:

Property

An interval is a set that consists of all the real numbers between two numbers $a$ and $b$.

1. The closed interval $[a, b]$ is the set $a \le x \le b$.
2. The open interval $(a, b)$ is the set $a < x < b$.
3. Intervals may also be half-open or half-closed, like $[a, b)$ which is $a \le x < b$.
4. The infinite interval $[a, \infty)$ is the set $x \ge a$.
5. The infinite interval $(-\infty, a]$ is the set $x \le a$.

A union of intervals, denoted with $\cup$, combines two or more sets.

Examples

• The inequality $-4 \le x < 1$ is written in interval notation as $[-4, 1)$.

• The set of all numbers greater than 5 is written as the infinite interval $(5, \infty)$.

Property

The absolute value of $x$ represents the distance from $x$ to the origin on the number line. Because distance is never negative, the absolute value is always non-negative. It is defined piecewise:

Examples

• To simplify $|3 - 8|$, first perform the operation inside the bars: $|3 - 8| = |-5|$. Then take the absolute value, which is $5$.
• In the expression $|3| - |8|$, we evaluate each absolute value first: $3 - 8$. The result is $-5$. This shows that $|a-b|$ is not always the same as $|a|-|b|$.
• To simplify $5 + 2|-4|$, we first evaluate the absolute value: $|-4| = 4$. The expression becomes $5 + 2(4) = 5 + 8 = 13$.

Explanation

Absolute value essentially makes any number positive. It measures a number's distance from zero on a number line, and distance is always a positive concept. Whether a number is positive or negative, its absolute value is its positive counterpart.