Writing Domain and Range in Interval Notation

You need to write the domain and range of a function using interval notation, and you keep getting the brackets wrong. The rule is simple: parentheses ( ) exclude the endpoint, brackets [ ] include it, and infinity always gets a parenthesis because you never reach infinity. This walks through the rules, the function types that trip students up, and the real mistakes that cost points on exams.

Three restrictions control most domain and range interval notation problems: division by zero, even roots that require non-negative radicands, and logarithms that demand positive arguments. Every other function type is either unrestricted or falls back on these three rules.

Division by Zero

A denominator cannot be zero. For f(x) = 1/(x − 3), set x − 3 = 0, find x = 3, then exclude that value. The domain is (−∞, 3) ∪ (3, ∞). OpenStax College Algebra 2e, Section 3.2, defines the domain this way for rational functions.

Even Roots

For square roots, fourth roots, and any even-index root, the radicand must be ≥ 0. For g(x) = √(5 − 2x), solve 5 − 2x ≥ 0 → x ≤ 2.5. The domain is (−∞, 2.5]. The bracket at 2.5 tells you the endpoint is included because the radicand can be zero.

Logarithms

The argument of a logarithm must be strictly greater than zero. For h(x) = log(3x + 1), set 3x + 1 > 0 → x > −1/3. The domain is (−1/3, ∞). The parenthesis at −1/3 means you cannot take the log of zero.

Each function family follows its own pattern for domain in interval notation. Memorize the pattern, not the numbers.

Rational Function

f(x) = (x + 1)/(x² − 4). Factor the denominator: (x − 2)(x + 2) = 0, so x = 2 and x = −2 are excluded. The domain is (−∞, −2) ∪ (−2, 2) ∪ (2, ∞). Two excluded values produce three intervals. The union symbol ∪ means "or", the input can be in any one of those intervals.

Radical Function

g(x) = √(4 − x²). Set 4 − x² ≥ 0 → x² ≤ 4 → −2 ≤ x ≤ 2. The domain is [−2, 2]. The brackets at both ends show that −2 and 2 are valid inputs because the radicand can be zero.

Logarithmic Function

h(x) = ln(2x − 5). Set 2x − 5 > 0 → x > 2.5.The parenthesis at 2.5 is non-negotiable, no log of zero.

Finding the range is harder because you are looking at outputs, not inputs. For common function families, the range is predictable.

Quadratic Function

f(x) = x² has vertex at (0,0) and opens upward. The smallest output is 0, there is no largest output. The range in interval notation is [0, ∞). OpenStax College Algebra 2e, Section 3.2, confirms this pattern for any quadratic with a positive leading coefficient.

Rational Function

f(x) = 1/x never outputs 0, and it approaches 0 from both sides but never reaches it. The range is (−∞, 0) ∪ (0, ∞). The gap at 0 is a vertical asymptote, not a hole.

Radical Function

g(x) = √x outputs only non-negative numbers. The range is [0, ∞). The bracket at 0 means the output can be exactly 0.

When a graph is given, the domain is the set of all x-values covered by the curve, and the range is the set of all y-values. Look at the horizontal extent for the domain, the vertical extent for the range. A filled dot means the endpoint is included (use a bracket); an open circle means the endpoint is excluded (use a parenthesis). If the graph extends forever in one direction, use ∞ or −∞ with a parenthesis. For example, a parabola opening upward with vertex at (1, −3) and extending both ways horizontally has domain (−∞, ∞) and range [−3, ∞). The bracket at −3 tells you the vertex output is included.

A piecewise function uses different rules for different parts of the domain. The domain is the union of the x-intervals where each piece applies. For f(x) = { x², x < 0; x + 1, x ≥ 0 }, the domain is (−∞, 0) ∪ [0, ∞) which simplifies to (−∞, ∞). The range requires evaluating each piece over its own interval and then taking the union. For the same function, the first piece x² on (−∞, 0) outputs (0, ∞), the second piece x + 1 on [0, ∞) outputs [1, ∞). The combined range is (0, ∞). The gap between 0 and 1 is not covered, no output falls between 0 and 1, so the range is an open interval starting at 0.

In calculus, you use interval notation to describe where a function is increasing or decreasing. A function is increasing on an interval if f(x₁) < f(x₂) whenever x₁ < x₂. For f(x) = x³ − 3x, the derivative f'(x) = 3x² − 3 gives critical points at x = −1 and x = 1. The function increases on (−∞, −1) and (1, ∞), and decreases on (−1, 1). The endpoints are critical points where the derivative is zero, and they are usually written with parentheses because the function is neither increasing nor decreasing at a flat point.

When facing a new function, take these steps in order. First, check for denominators: set each to zero and exclude the solutions. Second, check for even roots: set the radicand ≥ 0 and solve. Third, check for logs: set the argument > 0 and solve. Fourth, combine all conditions using the intersection of the intervals, the domain must satisfy every condition at once. For a function like f(x) = √(x − 1) / (x − 3), the radicand requires x ≥ 1, and the denominator excludes x = 3. The intersection is [1, 3) ∪ (3, ∞). Write it as a union of disjoint intervals. The failure case: writing [1, 3) ∪ (3, ∞] is wrong because infinity never gets a bracket.