Surjective functionIn mathematics, a surjective function (also known as surjection, or onto function) is a functionf such that, for every element y of the function's codomain, there exists at least one element x in the function's domain such that f(x) = y. In other words, for a function f : X → Y, the codomain Y is the image of the function's domain X. It is not required that x be unique; the function f may map one or more elements of X to the same element of Y. The term surjective and the related terms injective and bijective were introduced by Nicolas Bourbaki, a group of mainly French 20th-century mathematicians who, under this pseudonym, wrote a series of books presenting an exposition of modern advanced mathematics, beginning in 1935.
Domain of a functionIn mathematics, the domain of a function is the set of inputs accepted by the function. It is sometimes denoted by dom(f){\displaystyle \operatorname {dom} (f)} or domf{\displaystyle \operatorname {dom} f}, where f is the function. In layman's terms, the domain of a function can generally be thought of as "what x can be".
Multivalued functionIn mathematics, a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for at least one point in its domain. It is a set-valued function with additional properties depending on context; though some authors do not distinguish between set-valued functions and multifunctions.
Identity functionIn mathematics, an identity function, also called an identity relation, identity map or identity transformation, is a function that always returns the value that was used as its argument, unchanged. That is, when f{\displaystyle f} is the identity function, the equalityf(x)=x{\displaystyle f(x)=x} is true for all values of x{\displaystyle x} to which f{\displaystyle f} can be applied.
Multiplicative inverseIn mathematics, a multiplicative inverse or reciprocal for a numberx, denoted by 1x{\displaystyle {\tfrac {1}{x}}} or x, is a number which when multiplied byx yields the multiplicative identity, 1. The multiplicative inverse of a fractionab{\displaystyle {\tfrac {a}{b}}} is ba.{\displaystyle {\tfrac {b}{a}}.} Dividing 1 by a real number yields its multiplicative inverse.
Inverse trigonometric functionsInverse functions of sin, cos, tan, etc.In mathematics, the inverse trigonometric functions (occasionally also called antitrigonometric, cyclometric, or arcus functions) are the inverse functions of the trigonometric functions, under suitably restricted domains. Specifically, they are the inverses of the sine, cosine, tangent, cotangent, secant, and cosecant functions, and are used to obtain an angle from any of the angle's trigonometric ratios.
Involution (mathematics)In mathematics, an involution, involutory function, or self-inverse function is a functionf that is its own inverse, f(f(x)) = x for all x in the domain of f. Equivalently, applying f twice produces the original value. General propertiesAny involution is a bijection. The identity map is a trivial example of an involution.Function (mathematics)In mathematics, a function from a set X to a set Y assigns to each element of X exactly one element of Y. The set X is called the domain of the function and the set Y is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a function of time.
BijectionIn 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). Given a function f:A→B{\displaystyle f:A\to B}, the image of an elementa∈A{\displaystyle a\in A} is the element f(a)∈B{\displaystyle f(a)\in B} in the codomain.Real numberIn mathematics, a real number is a number that can be used to measure a continuous one-dimensionalquantity such as a length, duration or temperature. Here, continuous means that pairs of values can have arbitrarily small differences. Every real number can be almost uniquely represented by an infinite decimal expansion.
MathematicsMathematics is a field of knowledge concerned with abstract concepts such as numbers, geometric shapes, sets, functions, and probabilities. It uses logical reasoning and proof to study and establish their properties, often expressed as theorems, formulas, and equations. Mathematics is used to model and solve problems in science, engineering, technology, economics, and everyday life.
If and only ifIn logic and related fields such as mathematics and philosophy, "if and only if" (often shortened as "iff") states that the truth values of two statements are equal. It is paraphrased by the biconditional, a logical connective between statements. The biconditional is true in two cases, where either both statements are true or both are false.
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