A closed interval [a, b] is the set of all real numbers from a to b, including both endpoints, written with square brackets. In AP Precalculus, closed intervals show up when you restrict domains (like for inverse trig functions) and when you describe exactly where a function's values live.
A closed interval is a stretch of the number line that keeps both of its endpoints. Write it as [a, b], and it means every number x with a ≤ x ≤ b. The square brackets are the whole point. They tell you a and b are members of the set, not just boundaries you approach. Compare that to the open interval (a, b), which uses parentheses and throws the endpoints out.
In AP Precalculus, closed intervals are the language you use to describe domains, ranges, and behavior over a specific window. When you restrict the sine function to [-π/2, π/2] so it can have an inverse, that's a closed interval doing real work. When you say a function is increasing on [2, 5], or that the range of cos x is [-1, 1], you're using closed-interval notation. Getting the brackets right isn't a formality. [-1, 1] and (-1, 1) are different sets, and only one of them correctly says cosine actually reaches 1 and -1.
Closed intervals aren't a single CED topic; they're notation that runs through the whole course. In Unit 1, you describe where polynomial and rational functions are increasing, decreasing, positive, or negative, and you do it with interval notation. In Unit 3, the restricted domains that make inverse trig functions possible are closed intervals. Arcsine outputs values in [-π/2, π/2], and arccosine outputs values in [0, π]. The range of sine and cosine is the closed interval [-1, 1] precisely because both functions actually hit those endpoint values.
On the exam, sloppy brackets cost points. If a question asks for the range of f(x) = 3sin(x) and you write (-3, 3) instead of [-3, 3], you've claimed the function never reaches its max and min, which is wrong. Closed intervals are also how boundedness gets expressed. Saying a function is bounded means its outputs are trapped inside some interval, and when the function reaches its bounds, that interval is closed.
Keep studying AP Precalculus Unit 3
Open Interval (all units)
The flip side of the same idea. An open interval (a, b) excludes its endpoints, while [a, b] keeps them. You'll also mix them, like [0, 2π) for one full revolution around the unit circle, which includes 0 but stops just short of 2π so you don't count the same angle twice.
Domain and Range (Unit 1)
Interval notation is the standard way to write domains and ranges in AP Precalc. Whether the interval is closed or open depends on whether the function actually attains the boundary value. The range of x² is [0, ∞) because the function really does output 0.
Unit Circle and Inverse Trig (Unit 3)
Inverse trig functions only exist because we restrict sine, cosine, and tangent to intervals where they pass the horizontal line test. Arcsine lives on the closed interval [-π/2, π/2] and arccosine on [0, π]. Knowing these specific closed intervals is non-negotiable for evaluating inverse trig expressions correctly.
Bounded Function (Units 1 & 3)
A bounded function has outputs that stay inside an interval. Sine and cosine are the classic examples, with outputs locked in [-1, 1]. The interval is closed because both functions genuinely reach 1 and -1, which is why amplitude problems give exact max and min values.
You won't see a question that asks 'define closed interval,' but the notation is everywhere. Multiple-choice questions ask for the domain or range of a function, and the answer choices often differ only in brackets versus parentheses, so you have to decide whether the endpoint is actually included. Free-response questions ask you to describe where a function is increasing, decreasing, or concave up, and your answer needs correct interval notation. Inverse trig questions quietly test closed intervals too. If you evaluate arcsin(1/2) and give an answer outside [-π/2, π/2], it's wrong even if the sine value matches. The skill being tested is precision. Ask yourself one question every time you write an interval. Does the function actually reach this endpoint? If yes, bracket. If no, parenthesis.
A closed interval [a, b] includes both endpoints; an open interval (a, b) includes neither. The difference matters most at boundaries. The range of sin x is the closed interval [-1, 1] because sine actually equals 1 at π/2. But the domain of a rational function with a vertical asymptote at x = 3 must use parentheses around 3, since the function is undefined there. Quick rule. Bracket means the value belongs to the set, parenthesis means it's a boundary the set never touches. Infinity always gets a parenthesis because you can't 'include' infinity.
A closed interval [a, b] includes both endpoints, so it contains every x with a ≤ x ≤ b.
Square brackets mean the endpoint is included; parentheses mean it's excluded, and that single symbol can be the difference between a right and wrong answer.
The range of sine and cosine is the closed interval [-1, 1] because both functions actually reach 1 and -1.
Inverse trig functions are built on restricted closed domains, with arcsine outputting values in [-π/2, π/2] and arccosine in [0, π].
Infinity never gets a bracket, so intervals like [0, ∞) always end with a parenthesis on the infinite side.
Before writing any interval, check whether the function attains the endpoint value; if it does, use a bracket.
A closed interval [a, b] is the set of all real numbers from a to b, including both endpoints. You'll use it constantly to write domains, ranges, and intervals where a function is increasing or decreasing.
A closed interval [a, b] includes its endpoints; an open interval (a, b) excludes them. So [-1, 1] contains -1 and 1, while (-1, 1) contains everything between them but not the endpoints themselves.
No. Infinity is not a number you can include, so it always gets a parenthesis. The correct notation is [0, ∞), which is closed at 0 but open toward infinity. Writing a bracket next to ∞ is an automatic notation error.
Sine and cosine fail the horizontal line test on their full domains, so we restrict them to intervals where they're one-to-one. Those restrictions are closed because the functions attain their max and min there. Sine is restricted to [-π/2, π/2] and cosine to [0, π], and the inverse functions output values from those same intervals.
It's the closed interval [-1, 1]. Sine actually equals 1 at π/2 and -1 at -π/2, so both endpoints are attained and must be included with brackets. Writing (-1, 1) claims sine never reaches its peak, which is false.