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Domain



Key Terms
What is Domain
Expressing Domain
Natural Domain vs Restricted Domain
Finding Domain: Rational Functions
Finding Domain: Radical Functions
Finding Domain: Logarithmic Functions
Finding Domain: Trigonometric Functions
Finding Domain: Combined Functions
Domain from Graph
Domain from Context
Domain Rules at a Glance



Where Functions Live

Every function has boundaries. Some inputs work, others do not. The square root function refuses negative numbers. The reciprocal function cannot accept zero. A function modeling time accepts no moments before the clock starts.

The domain is the complete set of inputs for which the function produces valid output. It answers the question: what values can this function actually process? Finding the domain means identifying every restriction — every value that causes division by zero, every radicand that turns negative, every argument that falls outside the reach of the rule.

Key Terms



See All Function Definitions →


What is Domain

The domain of a function is the set of all inputs for which the function produces a valid output. It includes every value that can be substituted into the function and yield a real number result.

A function is not just a rule — it is a rule together with a domain. The expression f(x)=x2f(x) = x^2 describes a rule, but the function is not complete until the domain is specified. Often the domain is implicit: all real numbers for which the formula makes sense. Sometimes it is explicitly restricted to match a context or application.

The domain answers a simple question: what can go in? For f(x)=2x+5f(x) = 2x + 5, any real number works — there are no divisions, no roots, no restrictions. The domain is all real numbers, written (−∞,∞)(-\infty, \infty) in interval notation or R\mathbb{R} in set notation.

For f(x)=1/xf(x) = 1/x, the input x=0x = 0 fails because division by zero is undefined. Every other real number works. The domain is all real numbers except zero: (−∞,0)∪(0,∞)(-\infty, 0) \cup (0, \infty).

The domain is distinct from the range, which describes outputs rather than inputs. Together they frame the complete input-output picture of a function.

Expressing Domain

Domain can be expressed in several equivalent notations, each suited to different contexts.

Interval notation uses parentheses and brackets to describe continuous sets. A parenthesis excludes the endpoint; a bracket includes it. The interval [2,5)[2, 5) contains all numbers from 22 to 55, including 22 but excluding 55. The interval (−∞,3](-\infty, 3] contains all numbers up to and including 33. Infinity always takes a parenthesis because infinity is not a number that can be reached.

Set-builder notation describes the domain by a defining property. The set {x∣x≥0}\{x \mid x \geq 0\} reads "the set of all xx such that xx is greater than or equal to zero." This notation handles complex conditions that intervals cannot easily express.

Inequality notation states the condition directly: x≥0x \geq 0 or −3<x≤7-3 < x \leq 7. This form appears often in contexts where the domain restriction is being derived or explained.

Graphical representation shows the domain on a number line. A solid dot marks an included endpoint; an open dot marks an excluded one. Shading indicates the included values.

Disconnected domains use union notation. The domain (−∞,−1)∪(1,∞)(-\infty, -1) \cup (1, \infty) consists of all numbers less than −1-1 together with all numbers greater than 11, excluding everything in between.

The table below collects these notations side by side with their forms and representative examples.
Notation Form Example Note
Interval parentheses and brackets around endpoints [2, 5), (−∞, 3] bracket includes the endpoint; parenthesis excludes it; ∞ always takes a parenthesis
Set-builder {x | defining condition} {x | x ≥ 0} handles complex conditions intervals cannot easily express
Inequality condition stated directly on x x ≥ 0;   −3 < x ≤ 7 natural form when a restriction is being derived
Union (disconnected) two or more intervals joined with ∪ (−∞, −1) ∪ (1, ∞) used when the domain skips a value or interval

Natural Domain vs Restricted Domain

The natural domain of a function is the largest set of real numbers for which the formula produces real output. It is determined entirely by the algebraic expression — no external constraints, no contextual limitations.

For f(x)=x−3f(x) = \sqrt{x - 3}, the natural domain is [3,∞)[3, \infty) because the radicand x−3x - 3 must be non-negative. For g(x)=ln⁡(x)g(x) = \ln(x), the natural domain is (0,∞)(0, \infty) because logarithms require positive arguments. The natural domain follows from the mathematics alone.

A restricted domain is a deliberate limitation imposed beyond what the formula requires. The function f(x)=x2f(x) = x^2 has natural domain (−∞,∞)(-\infty, \infty), but restricting to [0,∞)[0, \infty) makes the function one-to-one, which is necessary for constructing an inverse.

Context often restricts domain. A function modeling the height of a projectile might be h(t)=−16t2+64th(t) = -16t^2 + 64t, which algebraically accepts any real tt. But if the projectile launches at t=0t = 0 and lands at t=4t = 4, the meaningful domain is [0,4][0, 4]. Negative time and post-landing time carry no physical significance.

The implied domain is the natural domain assumed when no domain is explicitly stated. Unless context or instruction specifies otherwise, assume the domain is as large as the formula allows.

Finding Domain: Rational Functions

A rational function is a ratio of two polynomials. The fundamental restriction is that the denominator cannot equal zero — division by zero is undefined.

To find the domain of a rational function, set the denominator equal to zero and solve. The solutions are the excluded values; everything else is in the domain.

For f(x)=x+3x−2f(x) = \dfrac{x + 3}{x - 2}, set x−2=0x - 2 = 0 to find x=2x = 2. The domain is all real numbers except 22:

(−∞,2)∪(2,∞)(-\infty, 2) \cup (2, \infty)


For g(x)=5x2−9g(x) = \dfrac{5}{x^2 - 9}, factor the denominator: x2−9=(x−3)(x+3)x^2 - 9 = (x-3)(x+3). Setting each factor to zero gives x=3x = 3 and x=−3x = -3. The domain is:

(−∞,−3)∪(−3,3)∪(3,∞)(-\infty, -3) \cup (-3, 3) \cup (3, \infty)


For h(x)=xx2+1h(x) = \dfrac{x}{x^2 + 1}, the denominator x2+1x^2 + 1 is always positive — it has no real roots. No value of xx makes it zero. The domain is all real numbers: (−∞,∞)(-\infty, \infty).

The numerator does not affect domain. A zero in the numerator makes the function equal zero, which is a valid output. Only zeros in the denominator create exclusions.

Finding Domain: Radical Functions

Radical functions involve roots — square roots, cube roots, fourth roots, and so on. The domain restrictions depend on whether the root index is even or odd.

Even roots (square root, fourth root, sixth root) require the radicand to be non-negative. Negative numbers have no real even roots. For f(x)=x−5f(x) = \sqrt{x - 5}, the radicand x−5x - 5 must satisfy x−5≥0x - 5 \geq 0, giving x≥5x \geq 5. The domain is [5,∞)[5, \infty).

Odd roots (cube root, fifth root) accept any real number. The cube root of −8-8 is −2-2, a perfectly valid real number. For g(x)=x−53g(x) = \sqrt[3]{x - 5}, no restriction applies. The domain is (−∞,∞)(-\infty, \infty).

More complex radicands require solving inequalities. For f(x)=6−2xf(x) = \sqrt{6 - 2x}, set 6−2x≥06 - 2x \geq 0:

6≥2x6 \geq 2x

3≥x3 \geq x


The domain is (−∞,3](-\infty, 3].

Nested radicals compound the restrictions. For f(x)=x−1f(x) = \sqrt{\sqrt{x} - 1}, both layers impose conditions. The inner radical requires x≥0x \geq 0. The outer radical requires x−1≥0\sqrt{x} - 1 \geq 0, which means x≥1\sqrt{x} \geq 1, so x≥1x \geq 1. The domain is [1,∞)[1, \infty).

Finding Domain: Logarithmic Functions

Logarithmic functions require their argument to be strictly positive. The logarithm of zero is undefined, and the logarithm of a negative number has no real value.

For f(x)=log⁡(x)f(x) = \log(x) or f(x)=ln⁡(x)f(x) = \ln(x), the domain is (0,∞)(0, \infty) — all positive real numbers.

When the argument is an expression, set that expression greater than zero and solve. For f(x)=ln⁡(x−4)f(x) = \ln(x - 4), require x−4>0x - 4 > 0:

x>4x > 4


The domain is (4,∞)(4, \infty).

For g(x)=log⁡(3−x)g(x) = \log(3 - x), require 3−x>03 - x > 0:

3>x3 > x


The domain is (−∞,3)(-\infty, 3).

Quadratic or factorable arguments require careful sign analysis. For h(x)=ln⁡(x2−4)h(x) = \ln(x^2 - 4), factor: x2−4=(x−2)(x+2)x^2 - 4 = (x-2)(x+2). This expression is positive when x<−2x < -2 or x>2x > 2. The domain is:

(−∞,−2)∪(2,∞)(-\infty, -2) \cup (2, \infty)


Note the strict inequality: unlike square roots, logarithms exclude the boundary where the argument equals zero.

Finding Domain: Trigonometric Functions

The basic trigonometric functions sine and cosine accept all real numbers — they have no domain restrictions. But tangent, cotangent, secant, and cosecant are defined as ratios, and ratios fail when the denominator is zero.

Tangent is defined as tan⁡(x)=sin⁡(x)/cos⁡(x)\tan(x) = \sin(x)/\cos(x). It is undefined where cos⁡(x)=0\cos(x) = 0, which occurs at x=π2+nπx = \dfrac{\pi}{2} + n\pi for any integer nn. The domain excludes these points.

Secant is sec⁡(x)=1/cos⁡(x)\sec(x) = 1/\cos(x), undefined at the same points as tangent.

Cotangent is cot⁡(x)=cos⁡(x)/sin⁡(x)\cot(x) = \cos(x)/\sin(x), undefined where sin⁡(x)=0\sin(x) = 0: at x=nπx = n\pi for any integer nn.

Cosecant is csc⁡(x)=1/sin⁡(x)\csc(x) = 1/\sin(x), undefined at the same points as cotangent.

Inverse trigonometric functions have restricted domains matching the ranges of the original functions. The function arcsin⁡(x)\arcsin(x) requires −1≤x≤1-1 \leq x \leq 1. The function arccos⁡(x)\arccos(x) has the same restriction. The function arctan⁡(x)\arctan(x) accepts all real numbers.

When trigonometric functions appear inside other operations, combine the restrictions. For f(x)=sin⁡(x)f(x) = \sqrt{\sin(x)}, the domain consists of all xx where sin⁡(x)≥0\sin(x) \geq 0 — the intervals where the sine function is non-negative.

The table below collects the six basic trigonometric functions with where each becomes undefined and the resulting domain exclusion.
Function Defined as Where undefined Domain
sin(x) — never all real numbers
cos(x) — never all real numbers
tan(x) sin(x) ⁄ cos(x) where cos(x) = 0 x ≠ π⁄2 + nπ
cot(x) cos(x) ⁄ sin(x) where sin(x) = 0 x ≠ nπ
sec(x) 1 ⁄ cos(x) where cos(x) = 0 x ≠ π⁄2 + nπ
csc(x) 1 ⁄ sin(x) where sin(x) = 0 x ≠ nπ

Finding Domain: Combined Functions

When multiple operations appear in a single function, every restriction applies simultaneously. The domain is the intersection of all individual requirements.

For f(x)=xx−3f(x) = \dfrac{\sqrt{x}}{x - 3}, two restrictions arise. The square root requires x≥0x \geq 0. The denominator requires x≠3x \neq 3. Both must hold, so the domain is [0,3)∪(3,∞)[0, 3) \cup (3, \infty).

For g(x)=ln⁡(x−1)+5−xg(x) = \ln(x - 1) + \sqrt{5 - x}, the logarithm requires x−1>0x - 1 > 0, giving x>1x > 1. The square root requires 5−x≥05 - x \geq 0, giving x≤5x \leq 5. The intersection is (1,5](1, 5].

For h(x)=1x−2h(x) = \dfrac{1}{\sqrt{x - 2}}, the square root requires x−2≥0x - 2 \geq 0, and the denominator requires x−2≠0\sqrt{x-2} \neq 0. Together these demand x−2>0x - 2 > 0, so x>2x > 2. The domain is (2,∞)(2, \infty).

A systematic approach prevents errors: list each operation that imposes a restriction, solve each restriction separately, then intersect all the resulting conditions. Overlapping constraints narrow the domain; contradictory constraints can eliminate it entirely.

Domain from Graph

When a function is given as a graph rather than a formula, the domain is read as the horizontal extent — the set of xx-values for which the graph exists.

Trace the graph from left to right. Where does it start? Where does it end? Does it continue indefinitely in either direction? The answers define the domain.

Solid dots indicate included endpoints. If the graph ends at a solid dot at x=3x = 3, then 33 is in the domain, and the interval is closed at that end: use a bracket.

Open dots indicate excluded endpoints. If the graph ends at an open dot at x=3x = 3, then 33 is not in the domain, and the interval is open at that end: use a parenthesis.

Vertical asymptotes signal domain exclusions. If the graph approaches but never touches a vertical line at x=2x = 2, then 22 is not in the domain. The domain splits into separate intervals on either side of the asymptote.

Gaps in the graph appear as breaks — intervals where no curve exists. A graph defined for x<−1x < -1 and x>1x > 1 but missing in between has domain (−∞,−1)∪(1,∞)(-\infty, -1) \cup (1, \infty).

Arrows at the ends of a graph indicate indefinite continuation. An arrow pointing right means the graph extends toward +∞+\infty; an arrow pointing left means it extends toward −∞-\infty.

The table below collects the visual cues that signal domain features on a graph and the notation each implies.
Visual cue What it means for the domain Notation impact
Solid dot at endpoint endpoint is included use bracket — [ or ]
Open dot at endpoint endpoint is excluded use parenthesis — ( or )
Vertical asymptote at x = a value a is excluded; graph approaches but never reaches domain splits at a using ∪
Gap in the graph interval missing — no curve exists there two or more disjoint intervals joined by ∪
Arrow at end of curve graph continues indefinitely in that direction extend to −∞ or +∞ with a parenthesis

Domain from Context

When a function models a real-world situation, the context imposes restrictions that pure algebra does not.

Time is typically non-negative. A function h(t)h(t) giving the height of a ball after tt seconds has no meaning for t<0t < 0. The domain starts at t=0t = 0, regardless of whether the formula accepts negative values.

Quantities like length, area, population, and count must be non-negative. A function A(r)=πr2A(r) = \pi r^2 giving the area of a circle requires r≥0r \geq 0, even though the formula works for negative rr.

Physical upper bounds also restrict domain. A function modeling the number of items produced in a day has a maximum determined by capacity. A function for temperature in a specific experiment operates within measured bounds.

Discrete quantities may restrict the domain to integers. A function P(n)P(n) giving profit based on nn units sold makes sense only for whole numbers n=0,1,2,3,…n = 0, 1, 2, 3, \ldots

Contextual domain is a modeling decision, not a calculation. It requires understanding what the variables represent and what values are meaningful in the situation. A formula may allow x=−5x = -5, but if xx represents the number of employees, that value is nonsense.

Domain Rules at a Glance

Finding the domain reduces to a small set of restriction rules — one or two per function type. The table below collects the rule for each function type covered above with a representative example and the resulting domain. It works as a study card and as a checklist when analyzing an unfamiliar expression: identify the operations involved, look up each restriction, then intersect the results.
Function type Restriction rule Example Resulting domain
Polynomial none f(x) = 2x + 5 (−∞, ∞)
Rational denominator ≠ 0 f(x) = 1 ⁄ (x − 2) (−∞, 2) ∪ (2, ∞)
Even radical radicand ≥ 0 f(x) = √(x − 5) [5, ∞)
Odd radical none f(x) = ∛(x − 5) (−∞, ∞)
Logarithmic argument > 0 f(x) = ln(x − 4) (4, ∞)
Trigonometric (tan, cot, sec, csc) underlying sin or cos ≠ 0 tan(x) x ≠ π⁄2 + nπ
Combined intersect every individual restriction f(x) = √x ⁄ (x − 3) [0, 3) ∪ (3, ∞)
Contextual / modeled constrained by physical meaning of the variable h(t) = −16t² + 64t (projectile) [0, 4]