Fact-checked by Grok 2 weeks ago

Domain of a function

In mathematics, the domain of a function refers to the complete set of all possible input values—often denoted as the set A in a function f: A \to B—for which the function produces a valid output, ensuring the expression defining the function is well-defined without encountering undefined operations such as division by zero or taking the square root of a negative number.^[1]^[2] This concept is fundamental in set theory, where a function is formally defined as a set of ordered pairs (x, y) with no two pairs sharing the same first element, and the domain consists precisely of all such first elements x.^[3] For real-valued functions commonly studied in calculus and analysis, the domain is typically a subset of the real numbers \mathbb{R}, determined by excluding points that lead to mathematical indeterminacies, and it plays a critical role in assessing properties like continuity, differentiability, and the function's overall behavior.^[4]^[5] The specification of a domain is essential when defining a function, as it delineates the scope of applicability; for instance, the natural domain of the rational function f(x) = \frac{1}{x} excludes x = 0 to avoid division by zero, resulting in the domain \mathbb{R} \setminus \{0\}.^[2] In broader contexts, such as multivariable functions or functions over abstract sets, the domain can be any mathematical structure—like vectors, matrices, or even other functions—provided the mapping adheres to the function's rules.^[3] Understanding the domain also contrasts with the related notions of codomain (the target set B) and range (the actual set of output values attained), which together fully characterize a function's structure and image.^[6] This framework underpins applications across fields like engineering, physics, and computer science, where precise domain restrictions ensure computational validity and model accuracy.^[7]

Definition and Fundamentals

Formal Definition

In mathematics, a function f: X \to Y from a set X to a set Y is formally defined as a relation that assigns to each element x in X exactly one element f(x) in Y. The domain of the function, denoted \dom(f) or simply the set X, is the set of all allowable input values for which the function is defined, ensuring that f(x) exists and lies within the codomain Y. This structure emphasizes that the function operates precisely on its specified domain, preventing undefined outputs.^[8] The domain delineates the scope of applicability of the function's rule, which can be described analytically, graphically, or via a formula, but always with the requirement of unique outputs for valid inputs. For instance, while the codomain Y represents the target set into which outputs are mapped, the domain X explicitly identifies where the mapping is valid, allowing for functions over arbitrary sets such as real numbers, integers, or more abstract structures.^[9] Functions are classified as total if their domain coincides with the entire intended input set, meaning the rule applies universally within that set, or partial if the domain is a proper subset, where the function is undefined for some potential inputs. This distinction is crucial in fields like analysis and computability, where partial functions model scenarios like division by zero or undecidable computations.^[10]

Notation and Terminology

In mathematical literature, the domain of a function f is frequently denoted using the symbol \dom(f) or D_f, where D represents the specific set comprising the domain.^[11] This notation emphasizes the domain as a distinct attribute of the function itself. Alternatively, functions are often specified in arrow notation as f: D \to Y, where D explicitly identifies the domain and Y the codomain, a convention rooted in set-theoretic definitions that clarifies the input-output structure.^[12] Terminology for the domain varies across contexts, with alternatives including "argument set" to highlight the collection of inputs or arguments accepted by the function, "input set" to stress the values fed into it, and "source" particularly in categorical or relational settings to denote the originating set in a mapping.^[13] The term "domain" originates from the Latin dominium, signifying ownership or lordship, which metaphorically captures the set over which the function asserts complete definitional control.^[14] This etymological root underscores the authoritative role of the domain in delimiting where the function is operational. The earliest documented use in a functional context appears in 1886, when Arthur Cayley employed it in his work on linear differential equations to describe the scope of variable applicability.^[15] Conventions for the domain differ by mathematical field: in analysis, it conventionally refers to a subset of the real numbers \mathbb{R} or \mathbb{R}^n, often with topological properties like openness or connectivity to support limits and continuity.^[11] In contrast, set theory treats the domain as an arbitrary set, without inherent structure beyond being a collection of elements that can be mapped, aligning with the foundational view of functions as relations between sets.^[16]

Domain in Real-Valued Functions

Natural Domain

In real analysis, the natural domain of a real-valued function is defined as the maximal subset of the real numbers on which the function's defining formula yields a real value, excluding points where operations such as division by zero or taking the logarithm of a non-positive number render the expression undefined.^[17] This set represents the largest possible domain inherent to the function's algebraic or transcendental expression without additional restrictions.^[18] For real analytic functions, expressed as power series, the natural domain corresponds to the interval or set of convergence of the series, serving as the initial domain from which analytic continuation can potentially extend the function to a larger set while preserving analyticity.^[19] This concept underscores the foundational role of the natural domain in identifying the primary region of validity before any such extensions are considered. The natural domain plays a crucial role in calculus by delineating the set over which properties like continuity and differentiability can be meaningfully analyzed, as these require the function to be defined and well-behaved at relevant points.^[20] Without specifying or respecting this domain, attempts to evaluate limits, derivatives, or integrals may lead to inconsistencies or undefined results. In contrast to explicitly specified domains, which may impose narrower subsets for modeling specific applications or constraints, the natural domain remains implicit and maximal, derived solely from the function's formula to encompass all feasible inputs.^[21] The notation \dom(f) is commonly used to denote this set.^[22]

Computing the Natural Domain

To compute the natural domain of a real-valued function, one systematically identifies all restrictions imposed by the operations within the function's expression, ensuring the output remains a real number.^[23] The process begins by examining each component for potential undefined behaviors, such as division by zero, even-powered roots of negative numbers, logarithms of non-positive numbers, and arguments outside the valid ranges for inverse trigonometric functions.^[24] These restrictions are derived from the fundamental properties of real arithmetic and transcendental functions, where operations like square roots require non-negative radicands to yield real results.^[25] A step-by-step approach involves first isolating each restrictive element. For denominators in rational expressions, set the denominator equal to zero and exclude those values from the real line, as division by zero is undefined in the reals.^[23] For radicals, particularly even roots like square roots, solve the inequality where the radicand is greater than or equal to zero; odd roots, such as cube roots, impose no such restriction on the argument but may require denominator checks if present.^[25] Logarithmic functions necessitate a positive argument, leading to inequalities like the input greater than zero.^[24] For inverse trigonometric functions, the domain is confined to the range of the corresponding trigonometric function over its principal branch: for example, \arcsin x and \arccos x require x \in [-1, 1], while \arctan x and \arccot x accept all real x, and \arccsc x and \arcsec x exclude [-1, 1] except the endpoints. Algebraic techniques center on solving the resulting inequalities to express the domain as intervals or unions thereof. Combine multiple constraints by intersecting the solution sets from each restriction, using methods like factoring, completing the square, or the rational root theorem for polynomial denominators and radicands.^[23] For instance, in a composite expression like \sqrt{x-2}/(x+1), the domain satisfies x \geq 2 from the radical and x \neq -1 from the denominator, yielding the intersection of these conditions.^[25] This intersection ensures all operations are valid simultaneously. For piecewise-defined functions, compute the natural domain of each piece separately, then take the union over the specified subintervals of the real line, excluding any points where a piece is undefined within its interval.^[26] The overall domain is thus the set of x values that fall into at least one valid piece and satisfy its internal restrictions. Graphically, the natural domain corresponds to the projection of the function's graph onto the x-axis, encompassing all horizontal coordinates where the curve is defined without vertical asymptotes, holes, or gaps arising from the restrictions.^[23] This visualization aids in verifying algebraic results, as undefined regions manifest as discontinuities or absences in the plot.

Examples and Applications

Polynomial and Rational Functions

Polynomial functions, being sums of powers of the variable with real coefficients, are defined for every real number input, resulting in a natural domain of all real numbers \mathbb{R}.^[27] This unrestricted domain stems from the fact that polynomials have no points of discontinuity or undefined behavior over the reals, allowing evaluation at any x \in \mathbb{R}.^[28] For instance, the quadratic polynomial f(x) = x^2 - 3x + 2 is well-defined for all real x, producing real outputs without exception.^[29] Rational functions, expressed as the ratio of two polynomials f(x) = \frac{p(x)}{q(x)} where q(x) is not the zero polynomial, have a natural domain consisting of all real numbers except those values where the denominator q(x) = 0.^[30] These exclusions arise because division by zero is undefined, creating gaps in the domain at the roots of q(x).^[31] A simple example is f(x) = \frac{1}{x-1}, whose domain is \mathbb{R} \setminus \{1\}, as the denominator vanishes at x = 1.^[32] To identify these domain exclusions, the denominator polynomial is typically factored to reveal its roots, using techniques such as the rational root theorem, synthetic division, or numerical root-finding methods.^[33] For a rational function like f(x) = \frac{x+2}{x^2 - 4}, factoring the denominator as (x-2)(x+2) shows exclusions at x = 2 and x = -2, after canceling the common factor x+2 (noting this creates a hole at x = -2 rather than a simple exclusion).^[34] At these boundary points where the denominator approaches zero (but the numerator does not), rational functions often exhibit vertical asymptotes, which visually and analytically indicate the domain gaps by showing unbounded behavior as x nears the excluded values.^[35] For f(x) = \frac{1}{x-1}, a vertical asymptote at x = 1 underscores the exclusion, with f(x) \to \pm \infty as x approaches 1 from either side.^[36]

Exponential and Logarithmic Functions

Exponential functions of the form f(x) = a^x, where the base a > 0 and a \neq 1, are defined for every real number x, resulting in a domain of all real numbers, \mathbb{R}. This unrestricted domain arises because exponentiation with a positive base extends naturally to all real exponents through limits and properties of real analysis, without encountering singularities or undefined points. For instance, the natural exponential function f(x) = e^x shares this full real domain, enabling its widespread use in modeling continuous growth processes.^[37]^[38] In contrast, logarithmic functions impose stricter domain requirements due to their inverse relationship with exponentials. The function f(x) = \log_b x, with base b > 0 and b \neq 1, is only defined for x > 0, yielding a natural domain of (0, \infty). This positivity constraint ensures the argument corresponds to the range of the corresponding exponential, avoiding non-positive inputs that would lack real-valued outputs. The base restrictions prevent degeneracy, as b = 1 yields a constant function and b \leq 0 introduces complex values or undefined behavior.^[39]^[40] When logarithmic functions are composed with other expressions, the domain depends on ensuring the inner function produces positive outputs. For example, in f(x) = \log_b (x^2 + 1), the argument x^2 + 1 \geq 1 > 0 holds for all real x, so the domain expands to \mathbb{R}, illustrating how algebraic adjustments can mitigate the inherent restrictions of logarithms. Such composites are common in applications like signal processing, where the full real line input is desirable.^[41]^[42] The change of base formula, \log_b x = \frac{\log_k x}{\log_k b} for any valid base k > 0, k \neq 1, preserves the original domain constraints without alteration, as both numerator and denominator inherit the positivity requirements from the logarithms involved. This equivalence facilitates numerical evaluation using computationally convenient bases like 10 or e, but the fundamental domain remains x > 0 and b > 0, b \neq 1.^[43]

Set-Theoretic and Abstract Perspectives

Domain as Projection of Relations

In set theory, a function f: X \to Y is formally defined as a subset of the Cartesian product X \times Y such that for every x \in X, there exists exactly one y \in Y with (x, y) \in f.^[44] This construction views functions as special binary relations where each element in the domain pairs uniquely with an element in the codomain.^[45] The domain of such a function, denoted \dom(f), is extracted as the projection of the relation onto the first coordinate:

\dom(f) = \{ x \in X \mid \exists y \in Y \text{ such that } (x, y) \in f \}.

This set comprises all first components of the ordered pairs in f, ensuring the domain aligns precisely with the inputs for which the function is defined.^[45] For total functions, where the relation covers all of X, this projection yields exactly X.^[44] This projection-based view extends naturally to arbitrary binary relations. For any relation R \subseteq A \times B, the domain is

\dom(R) = \{ a \in A \mid \exists b \in B \text{ such that } (a, b) \in R \},

capturing the set of elements from A that participate in at least one ordered pair in R, without requiring uniqueness.^[46] Unlike functions, relations may have elements in A with multiple or no pairings, so \dom(R) may be a proper subset of A.^[44] The axiom of choice plays a crucial role in guaranteeing the existence of functions with prescribed domains in set theory. It states that for any collection of nonempty sets \mathcal{H}, there exists a choice function f with domain \mathcal{H} such that f(H) \in H for every H \in \mathcal{H}.^[47] This ensures that, given a domain set and nonempty target sets for each element, a function realizing those selections can be constructed, underpinning many existence proofs in mathematics.^[47]

Partial Functions and Restrictions

In set theory and mathematics, a partial function from a set C to a set B is a binary relation that assigns to each element in some subset of C at most one element in B, where the domain is explicitly the subset on which it is defined, rather than all of C.^[48] This contrasts with total functions, which are defined on their entire specified domain; every total function is a partial function, but partial functions allow for undefined points within the formal domain.^[48] For instance, the function f: \mathbb{N} \to \mathbb{R} defined by f(n) = 1/n for n \geq 1 and undefined at n=0 (assuming \mathbb{N} includes 0) exemplifies a partial function, with domain \{n \in \mathbb{N} \mid n \geq 1\}.^[48] The restriction of a function f: X \to Y to a subset A \subseteq X, denoted f|_A: A \to Y, is the function that agrees with f on A, effectively narrowing the domain while preserving the mapping rule and codomain.^[49] This construction ensures that f|_A inherits the relational structure of f but operates only on the specified subdomain, useful for isolating behaviors or properties within subsets.^[49] Restrictions maintain certain inverse image properties, such as (f|_A)^{-1}(B) = A \cap f^{-1}(B) for B \subseteq Y.^[49] An extension of a function f: X \to Y is a function g: A \to B where X \subseteq A, Y \subseteq B, and g(x) = f(x) for all x \in X, thereby enlarging the domain while coinciding with the original on its domain.^[50] Extensions are not arbitrary but often constrained by continuity or analyticity requirements; for example, analytic continuation extends holomorphic functions beyond their initial domain while preserving values.^[50] This process builds on the relational projection of the domain, allowing incomplete mappings to be completed under specific conditions.^[50] Restrictions and extensions impact functional properties like injectivity and surjectivity. If f: X \to Y is injective, then its restriction f|_A for any A \subseteq X is also injective, as distinct elements in A map to distinct elements in Y by the original injectivity.^[51] However, surjectivity is not necessarily preserved under restriction: if f is surjective onto Y, f|_A may fail to cover all of Y if A excludes preimages of some elements. Extensions can alter these properties depending on how the additional domain points are mapped; for instance, an injective function may lose injectivity if the extension maps new points to existing images.^[52]

Advanced Contexts

Domain in Complex Analysis

In complex analysis, a domain is defined as a non-empty open connected subset of the complex plane ℂ, often required to be path-connected to ensure the existence of continuous paths between any two points within it.^[53]^[54] This topological structure is essential because holomorphic functions, which are complex differentiable in a neighborhood of every point, are typically defined and studied on such open connected sets.^[55] For instance, the function f(z) = \frac{1}{z} is holomorphic on the domain ℂ excluding the origin (z=0), where it has a simple pole, but analytic everywhere else in this punctured plane.^[55] Holomorphic functions are inherently local, but their global behavior on a domain reveals important properties like the maximum modulus principle or Cauchy's integral theorem, which rely on the connectivity of the domain.^[55] For multi-valued functions such as the complex logarithm \log(z), the natural domain excludes branch cuts to ensure single-valuedness and analyticity; the principal branch is typically defined on ℂ minus the non-positive real axis, where the argument is taken in (-\pi, \pi).^[56]^[55] Similarly, the square root function \sqrt{z} requires a branch cut along the negative real axis to define a holomorphic branch on ℂ excluding that ray, avoiding the branch point at z=0.^[56] To achieve full analyticity for multi-valued functions beyond these restricted domains in ℂ, Riemann surfaces provide a natural extension by constructing a multi-sheeted covering space where the function becomes single-valued and holomorphic.^[55] For example, the sine function \sin(z), defined by its power series \sum_{n=0}^{\infty} \frac{(-1)^n z^{2n+1}}{(2n+1)!}, is entire and holomorphic on the entire domain ℂ without restrictions.^[55] In contrast, functions like \sqrt{z} or \log(z) gain a complete holomorphic extension only on their associated Riemann surfaces, which resolve the multi-valuedness inherent in the complex plane.^[55]^[56]

Domain in Computer Science

In computer science, particularly within type theory and programming languages, the domain of a function refers to the set of valid input types or values that the function can accept, analogous to the mathematical concept but formalized through type signatures. For instance, in a function f: \mathbb{Z} \to \text{String}, the domain consists of all integers, ensuring that only inputs of type integer are permissible to produce a string output. This specification prevents mismatches during compilation or execution, as outlined in foundational type theories like Church's simple type theory, where function types \alpha \to \beta explicitly define the domain type \alpha as the carrier for arguments.^[57] Domain-specific languages (DSLs) extend this notion by tailoring functions to a particular problem domain, where the domain encompasses the specialized inputs and concepts relevant to that area, such as financial calculations or graphical layouts. In DSLs, function domains are constrained to domain-appropriate data, like stock prices for a trading DSL, allowing concise expressions that map domain-specific inputs to outputs without general-purpose overhead. This approach, as defined by Martin Fowler, focuses DSLs on limited expressiveness for a targeted domain to enhance productivity in function design and invocation. In database systems, the domain of an attribute denotes the set of allowable values for that attribute within a relational model, ensuring data integrity by restricting entries to predefined ranges or types. For example, the domain for an "age" attribute might be the integer interval [0, 150], excluding invalid values like negative numbers or excessively large figures. This concept originates from Edgar F. Codd's relational model, where domains define atomic values for attributes to maintain consistency across relations.^[58] Type checking in programming languages verifies that function inputs belong to the specified domain, thereby preventing runtime errors akin to mathematical undefined operations, such as applying a function outside its valid range. Static type checkers, performed at compile time, reject code with domain violations before execution, while dynamic checking at runtime catches remaining issues, both contributing to robust software by avoiding exceptions from type mismatches. This mechanism is central to type-safe languages, where domain adherence ensures predictable behavior and reduces error propagation.^[59]

References

[1]
Domain of a Function - Department of Mathematics at UTSA
Oct 21, 2021 · In mathematics, the domain or set of departure of a function is the set into which all of the input of the function is constrained to fall.
[2]
[PDF] DOMAIN OF FUNCTIONS - myweb
The domain of a function f, denoted Dom(f), is the set of all real numbers x such that f(x) is defined. • The above domain definition in set notation: ...
[3]
4.1 Definition and Examples
The set A is called the domain of f and the set B is called the codomain. We say two functions f and g are equal if they have the same ...
[4]
Algebra - The Definition of a Function - Pauls Online Math Notes
Jun 14, 2024 · From the definition the domain is the set of all x x 's that we can plug into a function and get back a real number. At this point, that means ...
[5]
3.1 What Are Functions? - MIT Mathematics
The first members of the pairs are called arguments and the whole set of them is called the domain of the function. Thus the arguments of f here are 1 , 2 1, 2 ...
[6]
Domain, Codomain, and Range - Ximera - Xronos
2 : An element is in the domain if... it is something that the function can compute to a number. it is something that makes sense in ...
[7]
Domain and Range | Engineering Math Resource Center
The domain represents the inputs to a system or a function. The range is the set of outputs that the system or function will produce when given the inputs to ...
[8]
[PDF] 5 Functions
We now give the formal definition of a function, which is motivated by the fact that all ... • The domain of a function f, written Domain(f), is the set of all ...
[9]
section 2.1
domain of a function. is taken to be the largest set consistent with the ... This section provides a formal definition of a function, gives examples ...
[10]
[PDF] Chapter 2 Relations, Functions, Partial Functions
When a partial function is a total function, we don't call it a “partial total function”, but simply a “func- tion”. The usual pratice is that the term ...
[11]
Lejeune Dirichlet - Biography
### Summary of Dirichlet's Contribution to Function Definition and Domain in 1837
[12]
Domain -- from Wolfram MathWorld
The meaning of "domain" in topology is a connected open set. Another meaning of "domain" is what is more properly known as an integral domain, i.e., a ring that ...
[13]
function - PlanetMath.org
Mar 22, 2013 · The triplet (f,A,B) ( f , A , B ) is usually written with the specialized notation f:A→B f : A → B . This notation visually conveys the fact ...Missing: mathematical | Show results with:mathematical<|separator|>
[14]
Functions - Discrete Mathematics - An Open Introduction
Function Definitions · A function is a rule that assigns each element of a set, called the domain , to exactly one element of a second set, called the codomain .
[15]
Unambiguous terminology for domains, ranges, sources and targets.
Jul 18, 2013 · No, there is no standard unambiguous terminology. The standard terms are "domain" and "range" but they are ambiguous between authors.Missing: alternative | Show results with:alternative
[16]
domain - Wiktionary, the free dictionary
Etymology. From Middle English demayne, demain (“rule”), from Old French demeine, demaine, demeigne ... Noun. domain m (plural domains). (Internet) domain.
[17]
origin of the terms "domain" and "range"?
Oct 14, 2017 · From Earliest Known Uses of Some of the Words of Mathematics: DOMAIN was used in 1886 by Arthur Cayley in “On Linear Differential Equations” in ...
[18]
[PDF] Sets and Functions - UC Davis Math
A set is a collection of objects, called the elements or members of the set. The objects could be anything (planets, squirrels, characters in Shakespeare's ...
[19]
Review of Functions
In this section, we provide a formal definition of a function and examine ... If x = 0 is in the domain of a function f , then f has exactly one y -intercept.
[20]
Function - Branko Curgus
Here we present the formal definition of a function. The definition from ... This domain is called the natural domain of a function. For example, the ...
[21]
[PDF] Analytic continuation of Dirichlet series with almost periodic ...
Classical examples of Taylor series with natural domain the unit disk ... For all Diophantine α and every real analytic periodic function g satisfying ...<|separator|>
[22]
[PDF] Define Domain In Mathematics
The importance of clearly identifying the domain cannot be overstated. It influences the behavior, continuity, differentiability, and integrability of functions ...
[23]
Formulas and Graphs - Ximera - The Ohio State University
May 17, 2025 · An applied domain is a subset of the natural domain. The applied domain includes all of the real numbers that make sense in the application.
[24]
[PDF] Chapter 10 Continuity
Dirichlet, in 1837, accepted Fourier's definition of a function and ... case the domain is understood to be the natural domain: the largest subset of R on.
[25]
Functions - Calculus I - Pauls Online Math Notes
Aug 13, 2025 · In simplest terms the domain of a function is the set of all values that can be plugged into a function and have the function exist and have a ...
[26]
[PDF] The Domain of a Function
Jul 9, 2018 · Definition: Domain of a function is the set of “input” or argument values for which the function is defined. How to find a domain of a given ...
[27]
https://ximera.osu.edu/csccmathematics/precalculus/precalculus/polynomialFunctions/polynomials
[28]
Piecewise Functions
The domain of the piecewise-defined function is the set of all values that: 1) satisfy one of its various conditions, and 2) produce some (real-valued) output. ...
[29]
Polynomials - Ximera - The Ohio State University
Oct 16, 2025 · Their domains include all real numbers. They are continuous everywhere. No discontinuities or singularities. No asymptotes on the graph. Their ...
[30]
[PDF] An Introduction to Real Analysis John K. Hunter - UC Davis Math
Every polynomial function is continuous on R and every rational function is continuous on its domain. Proof. The constant function f(x) = 1 and the identity ...
[31]
Section 1.1: Power Functions and Polynomials
Do all polynomial functions have as their domain all real numbers? Yes. Any real number is a valid input for a polynomial function. Using Factoring to Find ...
[32]
[PDF] Rational functions - Columbia Math Department
The domain of a rational function consists of all real numbers x except those for which the denominator is zero.
[33]
2-07 Asymptotes of Rational Functions
The domain of a rational function is all real numbers except those that cause the denominator to equal zero.
[34]
Domain of rational functions - Ximera - Xronos
In order to find the domain of this rational function we need to check two types of domain restrictions. The first, is any natural domain restrictions due to ...
[35]
Objective 1 - Domain of Rational Functions - Ximera - Xronos
To find the domain of a rational function, solve for when the denominator is equal to zero. If there are multiple fractions, remove when each denominator is ...
[36]
Let's Learn Domains of Rational Functions
Determining the domain of a rational function essentially boils down to determining the values of x that do not lie in the domain.
[37]
3.3 Graphs of Rational Functions - The Texas A&M University System
... vertical asymptotes or holes in the graph, if they exist. Per Theorem 3.2, vertical asymptotes and holes in the graph come from values excluded from the domain ...
[38]
[PDF] Limits Prep Worksheet 3 Rational Expression/Functions Rational ...
Another way a rational function may have a restricted domain will appear in the graph as a Vertical Asymptote. A vertical asymptote happens where a value of ...<|control11|><|separator|>
[39]
4.1: Exponential Functions - Mathematics LibreTexts
Apr 8, 2022 · ... domain of an exponential function is ( − ∞ , ∞ ). The base of an exponential function must be positive. The reason for this restriction ...
[40]
20.3: Evaluate and Graph Exponential Functions
Aug 13, 2020 · The natural exponential function is an exponential function whose base is e: f(x)=ex. The domain is (−∞,∞) and the range is (0,∞).
[41]
12.3: Logarithmic functions - Mathematics LibreTexts
Oct 3, 2021 · The domain of the logarithmic function is \(\{x|x > 0\}\) or \((0, ∞)\), i.e., the value (or argument) of the logarithm is always positive.
[42]
4.4: Logarithmic Functions - Mathematics LibreTexts
May 2, 2023 · the domain of the logarithm function with base b is ( 0 , ∞ ) . · the range of the logarithm function with base b is ( − ∞ , ∞ ) .
[43]
4.4: Graphs of Logarithmic Functions - Mathematics LibreTexts
Dec 21, 2020 · Before working with graphs, we will take a look at the domain (the set of input values) for which the logarithmic function is defined. Recall ...Finding the Domain of a... · Graphing Transformations of...
[44]
4.4: Graphs of Logarithmic Functions - Mathematics LibreTexts
Apr 30, 2022 · The domain is \((0,∞)\), the range is \((−∞,∞)\) and the \(y\)-axis is the vertical asymptote. We summarize these properties in the chart below.
[45]
Logarithm change of base rule intro (article) - Khan Academy
Learn how to rewrite any logarithm using logarithms with a different base. This is very useful for finding logarithms in the calculator!Missing: domain | Show results with:domain
[46]
https://scholar.harvard.edu/files/yxiang/files/ho4-set-function-2.pdf
[47]
Functions and Relations - CSC 208: Discrete Structures
A relation is a set of pairs. A function is a special case of a relation, defined as right-unique and left-total. Not all relations are functions.
[48]
[PDF] Set theory, relations, and functions (II)
Feb 1, 2017 · A relation is a set of ordered pairs. A function is a relation where each domain element pairs with only one range element.
[49]
The Axiom of Choice - Stanford Encyclopedia of Philosophy
Jan 8, 2008 · The Axiom of Choice states that, given any collection of mutually disjoint nonempty sets, it is possible to assemble a new set with one element ...
[50]
partial function - PlanetMath
Mar 22, 2013 · Clearly if f f is a function from A A to B B then it is a partial function from A A to B B , but a partial function need not be defined for ...Missing: mathematics | Show results with:mathematics<|control11|><|separator|>
[51]
restriction of a function - PlanetMath.org
Mar 22, 2013 · restriction of a function. Definition. Let f:X→Y f : X → Y be a function from a set X X ...Missing: mathematics | Show results with:mathematics
[52]
extension of a function - PlanetMath.org
Mar 22, 2013 · extension of a function ... of g g to X X . Typically, functions are not arbitrarily extended. Rather, it is usually insisted upon that extensions ...
[53]
Injective, surjective and bijective functions - SIUE
Determine whether or not the restriction of an injective function is injective. If it is, prove your result. If it isn't, provide a counterexample. As we ...
[54]
Properties of Functions - Department of Mathematics at UTSA
Jan 11, 2022 · An extension of a function f is a function g such that f is a restriction of g. A typical use of this concept is the process of analytic ...
[55]
[PDF] Part IB - Complex Analysis (Definitions) - Dexter Chua
Definition (Domain). A domain is a non-empty open path-connected subset of C. Definition (Differentiable function). Let U ⊆ C be a domain and f : U → C be ...
[56]
2.2 Domains
A set D ⊂ C is a domain if it is non-empty, open and path connected. Throughout the course we will be working with domains.
[57]
[PDF] Lectures in Complex Analysis Dana Berman - McGill University
Feb 17, 2019 · Another interesting fact is that if f and g are holomorphic and equal on some region, then f = g on the entire complex plane. All of the above ...
[58]
[PDF] Branch Points and Branch Cuts (18.04, MIT). - MIT Mathematics
Oct 11, 1999 · Thus the origin is a branch point of log(z). Definition 1.1 The point z0 is called a branch point | for the complex (multiple) valued function f ...
[59]
Church's type theory - Stanford Encyclopedia of Philosophy
Aug 25, 2006 · A set or property can be represented by a function (often called characteristic function) which maps elements to truth values, so that an ...
[60]
[PDF] A Relational Model of Data for Large Shared Data Banks
Normally, one domain (or combination of domains) of a given relation has values which uniquely identify each ele- ment (n-tuple) of that relation. Such a domain ...
[61]
Type Safety in Programming Languages - Baeldung
Mar 26, 2025 · Type safety in programming languages enables the language to avoid type errors by ensuring variables access only authorized memory locations.