Functions and mappings

association of a single output to each input

thumb|300px|A definite integral of a function can be represented as the signed area of the region bounded by its graph and the horizontal axis; in the above graph as an example, the integral of f(x) between a and b is the yellow (−) area subtracted from the blue (+) area|alt=Definite integral example

In mathematics, the derivative is a fundamental tool that quantifies the sensitivity to change of a function's output with respect to its input. The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. The derivative is often described as the instantaneous rate of change, the ratio of the instantaneous change in the dependent variable to that of the independent variable. The process of finding a d

representation of a function as the set of pairs (x, f(x))

In mathematics, a bijection, bijective function, or one-to-one correspondence is a function between two sets such that each element of the second set (the codomain) is the image of exactly one element of the first set (the domain). Equivalently, a bijection is a relation between two sets such that each element of either set is paired with exactly one element of the other set.

mathematical function that preserves distinctness

form of graphical projection where the projection lines converge to one or more points

assignment of a vector to each point in a subset of Euclidean space

function such that every element of the codomain has a preimage

mapping that preserves the operations of addition and scalar multiplication

set of "input" or argument values for which a function is defined

In mathematics and more specifically in topology, a homeomorphism (from Greek roots meaning "similar shape", named by Henri Poincaré), also called topological isomorphism, or bicontinuous function, is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same.

operation which takes two mathematical functions and makes one function of these

point to which functions converge in analysis

element of the domain where function's value is zero

assignment of numbers to points in space

function that always returns the same value that was used as its argument

in Euclidean geometry, a function that moves every point a constant distance in a specified direction

right|thumb|250px|A function from to . The blue oval is the codomain of . The yellow oval inside is the Image (mathematics)|image of , and the red oval is the domain of .

thumb|upright=1.4|A Function composition|composition of two opposite isometries is a direct isometry. A reflection in a line is an opposite isometry, like (reflection w.r.t the center diagonal line) or (reflection w.r.t the right diagonal line) on the image. Translation is a direct isometry: a rigid motion.

function that is its own inverse

calculus theorem that the limit of a function trapped between two other functions with the same limit 𝐿 is also 𝐿

any function whose domain is the positive integers and whose range is a subset of the complex numbers

function from a set having some geometric structure to itself or another such set

mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points

analytic function that does not satisfy a polynomial equation

function which is defined by multiple sub-functions each over its own interval

function, sometimes assumed structure-preserving in a proper sense

binary relation, which is left-total, but may not be right-unique ; isomorph to another function from the same source set, but to the power set of the codomain of the initial function

mathematical operation on Euclidian spaces which reverses distances with respect to a given point

function whose actual domain of definition may be smaller than its input set

type of isometry of Euclidean space

mathematical operation

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selection of some objects in a particular order

function mapping a set to itself

assignment of a tensor continuously varying across a mathematical space

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In mathematics, an embedding (or imbedding) is one instance of some mathematical structure contained within another instance, such as a group that is a subgroup.

methods in computer graphics to project three-dimensional objects onto a plane by means of numerical calculations

fractional linear transformation on the complex projective line

function that maps a k-element real-valued vector to a k-element categorical probability distribution

ambiguous term, referring either to the codomain or the image of a function

arithmetic function

thumb|300px|Graph y=f(x) with the x-axis as the horizontal axis and the y-axis as the vertical axis. The y-intercept of f(x) is indicated by the red dot at (x=0, y=1). In analytic geometry, using the common convention that the horizontal axis represents a variable x and the vertical axis represents a variable y, a y-intercept or vertical intercept is a point where the graph of a function or relation intersects the y-axis of the coordinate system.

particular type of mapping in linear algebra, also called transvection

function that can be computed with loops of bounded length

limit of a function approaching a value point from values below or above the value point

mathematical operation of composing a function with itself repeatedly

function used as a performance test problem for optimization algorithms

or inclusion function, or canonical injection

concept in linear algebra

mathematical concept

function which encodes two natural numbers into a single natural number

expression in propositional calculus

computational problem where a single output (of a total function) is expected for every input, but the output is more complex than that of a decision problem

in hyperbolic geometry, the angle at one vertex of a right hyperbolic triangle that has two hyperparallel sides

bijection of an Euclidean space that preserves distances

mathematical function revertible near each point

A posynomial, also known as a posinomial in some literature, is a function of the form