How to Write the Domain of a Function in Interval Notation
Understanding the domain of a function is a fundamental skill in algebra, calculus, and many applied fields. Once you have identified that set, expressing it in interval notation provides a concise, standardized way to communicate the result. The domain tells you the set of all input values (usually x) for which the function produces a real‑valued output. This article walks you through the concept, the step‑by‑step process, common function types, and plenty of examples so you can confidently write any domain in interval notation.
What Is the Domain of a Function?
The domain of a function f is the collection of all real numbers x that can be substituted into the function’s formula without causing an undefined operation (such as division by zero, taking the square root of a negative number, or evaluating a logarithm of a non‑positive argument). In symbols, if f: D → ℝ, then D is the domain.
Interval notation is a shorthand that uses brackets and parentheses to describe subsets of the real line:
- [a, b] means all x such that a ≤ x ≤ b (closed interval, includes endpoints).
- (a, b) means all x such that a < x < b (open interval, excludes endpoints).
- [a, b) or (a, b] are half‑open intervals, including one endpoint but not the other.
- (−∞, b) or (a, ∞) extend indefinitely in one direction; infinity is always paired with a parenthesis because it is not a real number that can be “reached”.
General Procedure for Finding the Domain
Follow these steps for almost any algebraic expression:
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Identify restrictions
Look for operations that impose limits:- Denominators cannot be zero.
- Even‑root radicands (square root, fourth root, etc.) must be ≥ 0.
- Logarithmic arguments must be > 0.
- Inside a tangent, cotangent, secant, or cosecant function, avoid points where the cosine or sine is zero, respectively.
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Solve each restriction
Set up inequalities or equations that capture the forbidden values, then solve for x. -
Combine the allowable intervals
The domain is the set of real numbers that satisfy all restrictions simultaneously. Use intersection (∩) of the individual allowed sets Not complicated — just consistent.. -
Write the result in interval notation
Convert the combined set into one or more intervals, using the appropriate brackets/parentheses. -
Check for isolated points
Rarely, a function may be defined only at a single point (e.g., f(x) = √(x‑3) / √(x‑3) simplifies to 1 except at x = 3 where it is 0/0, undefined). In such cases, the domain may be a union of intervals plus isolated points; however, most elementary functions yield only intervals.
Examples by Function Type
1. Polynomial Functions
Example: f(x) = 2x⁴ − 3x² + 5
Polynomials involve only addition, subtraction, multiplication, and non‑negative integer powers of x. No division, roots, or logs appear, so every real number is allowed.
Domain: (−∞, ∞) or simply ℝ Most people skip this — try not to..
2. Rational Functions
Example: f(x) = (3x + 1) / (x² − 4)
Restriction: denominator ≠ 0 → x² − 4 ≠ 0 → (x − 2)(x + 2) ≠ 0 → x ≠ 2 and x ≠ −2 Turns out it matters..
Domain: (−∞, −2) ∪ (−2, 2) ∪ (2, ∞).
3. Even‑Root Functions (Square Root, Fourth Root, …)
Example: f(x) = √(5 − 2x)
Restriction: radicand ≥ 0 → 5 − 2x ≥ 0 → −2x ≥ −5 → x ≤ 5/2.
Domain: (−∞, 5/2].
Example with a fraction inside the root: f(x) = √[(x + 3)/(x − 1)]
Two steps:
- Test intervals: (−∞, −3] gives positive, (−3, 1) gives negative, (1, ∞) gives positive. Solve (x + 3)/(x − 1) ≥ 0 using a sign chart. Practically speaking, 2. The whole fraction must be ≥ 0. Critical points at x = −3 and x = 1. That's why denominator of the fraction inside the root cannot be zero → x ≠ 1. Include x = −3 because the fraction equals zero (root of zero is defined). Exclude x = 1.
Domain: (−∞, −3] ∪ (1, ∞).
4. Logarithmic Functions
Example: f(x) = ln(2x − 7)
Restriction: argument > 0 → 2x − 7 > 0 → 2x > 7 → x > 7/2 But it adds up..
Domain: (7/2, ∞).
Example with a quadratic inside: f(x) = log₃(x² − 4x + 3)
Set x² − 4x + 3 > 0 → (x − 1)(x − 3) > 0. Here's the thing — test intervals: (−∞, 1) positive, (1, 3) negative, (3, ∞) positive. Critical points at x = 1, 3. Endpoints excluded because log of zero is undefined Practical, not theoretical..
You'll probably want to bookmark this section It's one of those things that adds up..
Domain: (−∞, 1) ∪ (3, ∞) And that's really what it comes down to..
5. Trigonometric Functions
Example: f(x) = tan(x)
Recall tan x = sin x / cos x. Undefined where cos x = 0 → x = π/2 + kπ, k ∈ ℤ.
Domain: All real numbers except those points. In interval notation, you can describe one repeating
5. Trigonometric Functions (continued)
Example: (f(x)=\tan x)
The tangent function is defined wherever its denominator (\cos x) is non‑zero. On the flip side, ] Thus the domain consists of all real numbers except those points. (\cos x=0) at [ x=\frac{\pi}{2}+k\pi,\qquad k\in\mathbb Z . In interval notation one can describe the pattern as a union of open intervals that repeat every (\pi) units: [ \bigcup_{k\in\mathbb Z}\left(\frac{\pi}{2}+k\pi,;\frac{3\pi}{2}+k\pi\right) ] which is equivalent to (\displaystyle \mathbb R\setminus\Big{\frac{\pi}{2}+k\pi\mid k\in\mathbb Z\Big}) Small thing, real impact. Nothing fancy..
6. Absolute‑Value Functions
Example: (f(x)=|x^{2}-5x+6|)
Absolute value imposes no new restriction; the expression inside can be any real number. Consequently the domain is all real numbers: [ \text{Domain}=(-\infty,\infty). ]
Example with a denominator: (f(x)=\frac{|2x-3|}{x-1})
Here the denominator still cannot be zero, while the absolute value is harmless. Hence [ x\neq1\quad\Longrightarrow\quad\text{Domain}=(-\infty,1)\cup(1,\infty). ]
7. Exponential Functions
Example: (f(x)=5^{,x}) (or any (a^{x}) with (a>0,;a\neq1))
The exponent can be any real number, so the domain is the entire real line: [ \text{Domain}=(-\infty,\infty). ]
Example with a shift: (f(x)=2^{,3x+2})
Again no restriction, giving the same full‑line domain And that's really what it comes down to..
8. Composite Functions
When a function is built by composing two or more elementary functions, the domain is the set of inputs that satisfy all inner‑function restrictions and any restrictions imposed by the outer function.
Example: (f(x)=\ln!\bigl(\sqrt{x+4},\bigr))
- The square‑root requires (x+4\ge0;\Rightarrow;x\ge-4).
- The natural log needs a positive argument, so (\sqrt{x+4}>0). This holds for every (x>-4) (the endpoint gives (\sqrt{0}=0), which is not allowed).
Combining the two conditions: [ \text{Domain}=(-4,\infty). ]
Example with a rational inside a log: (f(x)=\log_{2}!\Bigl(\frac{x-1}{x+3}\Bigr))
- Denominator of the fraction: (x+3\neq0;\Rightarrow;x\neq-3).
- Log argument must be positive: [ \frac{x-1}{x+3}>0. ] Critical points are (x=1) and (x=-3). A sign chart shows the fraction is positive on ((-\infty,-3)\cup(1,\infty)).
Intersecting with the exclusion of (-3): [ \text{Domain}=(-\infty,-3)\cup(1,\infty). ]
9. Piecewise‑Defined Functions
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