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Home > Function (mathematics)

4 Domains, codomains, and ranges

X, the set of input values, is called the domain of f, and Y, the set of possible output values, is called the codomain. The range of f is the set of all actual outputs {f(x) : x in the domain}. Beware that sometimes the codomain is incorrectly called the range because of a failure to distinguish between possible and actual values.

Functions are named after their ranges, for example real functions and complex functions.

An endofunction is a function whose domain and range are identical.

In computer science, the datatypes of the arguments and return values specify the domain and codomain (respectively) of a subprogram. So the domain and codomain are constraints imposed initially on a function; on the other hand the range has to do with how things turn out in practice.

5 Injective, surjective and bijective functions

Several types of functions that are very useful have special names:

• Injective (one-to-one) functions send different arguments to different values; in other words, if x and y are members of the domain of f, then f(x) = f(y) only if x = y.
• Surjective (onto) functions have their range equal to their codomain; in other words, if y is any member of the codomain of f, then there exists at least one x such that f(x) = y.
• Bijective functions are both injective and surjective; they are often used to show that the sets X and Y are the "same size" in some sense.

6 Images and preimages

The image of an element x∈X under f is the output f(x).

The image of a subset A⊂X under f is the subset of Y defined by

f(A) = {f(x) | x in A}

Notice that the range of f is the image f(X) of its domain. In our function above, the image of {2,3} under f is f({2, 3}) = {c, d} and the range of f is {a, c, d}.

The preimage (or inverse image) of a set B ⊂ Y under f is the subset of X defined by

f ⁻¹(B) = {x in X | f(x) ∈B}

For our function above, the preimage of {a, b} is f ⁻¹({a, b}) = {1}.

7 Graph of a function

The graph of a function f is the set of all ordered pairs(x, f(x)), for all x in the domain X. There are theorems formulated or proved most easily in terms of the graph, such as the closed graph theorem. If X and Y are real lines, then this definition coincides with the familiar sense of graph.

the graph of a cubic function, This function is surjective but not injective.

Note that since a relation on the two sets X and Y is usually formalized as a subset of X×Y, the formal definition of function actually identifies the function f with its graph.

8 Examples of functions

(More can be found at List of functions.)

• The relation wght between persons in the United States and their weights at a particular time.
• The relation between nations and their capitals, if we exclude those nations that maintain multiple capitals [1].
• The relation sqr between natural numbers n and their squares n².
• The relation ln between positive real numbers x and their natural logarithms ln(x). Note that the relation between real numbers and their natural logarithms is not a function because not every real number has a natural logarithm; that is, this relation is not total.
• The relation dist between points in the plane R² and their distances from the origin (0,0).
• The relation grav between a point in the punctured plane R² \ {(0,0)} and the vector describing the gravitational force that a certain mass at that point would experience from a certain other mass at the origin (0,0).

The most commonly used types of mathematical functions involve addition, division, exponents, logarithms, multiplication, polynomials, radicals, rationals, subtraction, and trigonometric expressions. They are sometimes collectively referred as elementary functions -- but the meaning of this term varies among different branches of mathematics. Example of non-elementary functions (or special functions) are Bessel functions and gamma functions.

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Topics: Function (mathematics) Functions Relation Output Set Input...

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