An interval of a function is a continuous range of numbers along the number line, used to describe where a function is defined, where it produces output, or where it behaves in a specific way (increasing, decreasing, positive, or negative). In practical terms, when someone refers to “the interval of a function,” they’re usually talking about the function’s domain or range written in interval notation, or they’re identifying specific stretches of the graph where the function does something notable.
What an Interval Actually Is
An interval is a set of real numbers that includes every number between two endpoints. For example, all the numbers between 0 and 1 form an interval. So do all numbers greater than 5, or all numbers less than or equal to 10. The key idea is that there are no gaps: if two numbers are in the interval, every number between them is also in the interval.
Intervals show up constantly when working with functions because functions operate on continuous stretches of numbers. The set of all valid inputs (the domain) is expressed as one or more intervals. The set of all possible outputs (the range) is also expressed as intervals. And when you’re asked to identify where a function is increasing or where it dips below the x-axis, you answer with intervals.
How Interval Notation Works
Interval notation uses brackets and parentheses to show exactly which numbers are included in a set. The smaller number always comes first, followed by a comma, then the larger number. The type of bracket on each side tells you whether the endpoint itself is part of the set.
- Square bracket [ or ] means the endpoint is included. Think of it as a closed door: the boundary value is inside.
- Parenthesis ( or ) means the endpoint is excluded. The values get infinitely close to that number but never reach it.
So [3, 5) means all numbers from 3 up to (but not including) 5. The number 3 is in the set, 4.999 is in the set, but 5 is not. The interval (2, 7) includes everything strictly between 2 and 7, excluding both endpoints. And [1, 10] includes 1, 10, and every number in between.
Types of Intervals
There are a few categories worth knowing, since they come up repeatedly in algebra, precalculus, and calculus.
Open intervals like (a, b) exclude both endpoints. Neither a nor b is part of the set. Closed intervals like [a, b] include both endpoints. Half-open intervals (also called half-closed) include one endpoint but not the other: [a, b) includes a but not b, while (a, b] includes b but not a.
Intervals can also extend infinitely in one or both directions. The interval (3, ∞) means all numbers greater than 3, stretching out forever to the right. The interval (−∞, 5] means all numbers less than or equal to 5, extending endlessly to the left. And (−∞, ∞) represents every real number. Infinity always gets a parenthesis, never a bracket, because infinity isn’t an actual number you can reach or include.
Describing Domain and Range With Intervals
The most common reason you’ll use intervals in a function context is to express domain (all valid inputs) and range (all possible outputs). A simple function like f(x) = x² has a domain of (−∞, ∞) because you can square any real number. Its range, however, is [0, ∞) because squaring a number never produces a negative result, and it does hit zero when x is 0.
Certain types of functions naturally restrict their domains. A square root function like f(x) = √(x − 1) requires the expression under the root to be zero or positive, so x must be 1 or greater. The domain is [1, ∞). A function with a fraction, like f(x) = 1/x, can’t allow the denominator to equal zero, so its domain is every real number except 0. You’d write that as two intervals joined together: (−∞, 0) ∪ (0, ∞).
That ∪ symbol is the union sign. It combines separate intervals into a single description. Whenever a function’s domain or range has a gap, you use a union to connect the pieces. For instance, the function f(x) = x/(x² − 3x + 2) is undefined at x = 1 and x = 2 because those values make the denominator zero. Its domain would be written as (−∞, 1) ∪ (1, 2) ∪ (2, ∞), three separate intervals joined by unions.
Intervals Where a Function Increases or Decreases
Beyond domain and range, you’ll often be asked to find intervals where a function is increasing, decreasing, positive, or negative. These describe the behavior of the function across different stretches of its graph.
A function is increasing on an interval when its output values rise as you move from left to right. It’s decreasing when the outputs fall. If a function’s graph goes up until x = 4, then turns and goes down until x = 9, you’d say the function is increasing on (−∞, 4) and decreasing on (4, 9). Notice that the turning point itself is typically excluded from both intervals, because at that exact spot the function is neither rising nor falling.
Similarly, a function is positive on intervals where its graph sits above the x-axis, and negative where the graph sits below. If a function crosses the x-axis at x = 2 and x = 7, and the graph is above the axis between those points, the function is positive on (2, 7). At x = 2 and x = 7 the function equals zero, so those points aren’t included in the positive or negative intervals.
Reading Intervals From a Graph
When you’re looking at a graph, the endpoints of a function’s domain are marked with either a filled (solid) dot or an open (hollow) dot. A filled dot means the endpoint is included, corresponding to a square bracket in interval notation. An open dot means the endpoint is excluded, corresponding to a parenthesis. If the graph has an arrow extending off the edge, the function continues toward infinity in that direction.
For example, if a graph starts with a filled dot at x = −2, curves through several values, and ends with an open dot at x = 5, the domain is [−2, 5). If the same graph dips to a lowest y-value of 1 (with a filled dot on the curve at that height) and peaks at y = 8 (also with a filled dot), the range is [1, 8].
Interval Notation vs. Set-Builder Notation
You’ll sometimes see the same information written in set-builder notation instead of interval notation. Set-builder notation uses a description inside curly braces: {x | 4 < x < 8} reads as “the set of all x such that x is between 4 and 8.” That’s the same set as the interval (4, 8).
Both notations communicate identical information. Interval notation is more compact and is the standard in most algebra and calculus courses. Set-builder notation is more flexible when describing sets that don’t fit neatly into intervals, like “all integers” or “all x where x is not equal to 3.” In practice, you’ll see interval notation far more often when working with functions, especially for domain, range, and behavioral intervals.