Interval Notation

Definition

Interval notation is a concise way to represent the domain and range of functions. It specifies sets of numbers using intervals, which are essentially ranges of values. Here's how it works for domain and range:

Domain

The domain is the set of all possible input values (x) for which a function is defined. Interval notation for the domain describes these input values as intervals on the number line.

Common Interval Notations:

• (a, b): All values x such that a < x < b. The interval is open, meaning a and b are not included.

• [a, b]: All values x such that a < x < b. The interval is closed, meaning a and b are included.

• (a, b]: All values x such that a < x < b. The interval is right-closed and left-open.

• [a, b): All values x such that a < x < b. The interval is left-closed and right-open.

• (-∞, b): All values x such that x < b. The interval extends to negative infinity and b is not included.

• (-∞, b]: All values x such that x < b. The interval extends to negative infinity and b is included.

• [a, ∞): All values x such that x > a. The interval extends to positive infinity and a is included.

• (a, ∞): All values x such that x > a. The interval extends to positive infinity and a is not included.

Range

The range is the set of all possible output values (y) that a function can produce. Interval notation for the range describes these output values in a similar manner to the domain.

Examples:

• If a function has a domain of all real numbers from -2 to 5, the domain in interval notation is [-2, 5].

• If the range of a function is all values greater than or equal to 3, the range in interval notation is [3, ∞).

• If the domain is all real numbers except -1, the domain can be written as (-∞, -1)(-1, ∞), where  denotes the union of intervals.

In summary, interval notation is a streamlined way to describe sets of numbers within a specific range, providing clarity in defining both domains and ranges for functions.