# function

A function is a way of expressing the *dependence* of one quantity on another quantity
or quantities. Traditionally, functions were specified as explicit rules
or formulas that converted some input value (or values) into an output value.
If *f* is the name of the function and *x* is a name for an input
value, then *f*(*x*) denotes the output value corresponding to *x* under the rule *f*. An input value is also called an **argument** of the function, and an output value is called a *value* of the function.
The graph of the function *f* is the collection of all pairs (*x*, *f*(*x*)), where *x* is an argument of *f*.

For example, the circumference *C* of a circle depends on its diameter *d* according to the formula *C* = π*d*; therefore, one
can say that the circumference is a function of the diameter, and the functional
relationship is given by *C*(*d*) = π*d*. Equally well,
the diameter can be considered a function of the circumference, with the
relationship given by *d*(*C*) = *d*/π.

Again, if *y* = *ax*^{2} + *bx* + *c*,
where *a*, *b*, and *c* are constants, and *x* and *y* are variables, then *y* is said to be a function of *x*, since, by assigning to *x* a series of different values,
a corresponding series of values of *y* is obtained, showing its
dependence on the value given to *x*. For this reason, *x* is termed the independent variable and y the dependent variable.

There may be more than one independent variable – e.g. the area of a triangle depends on its altitude and its base, and is thus a function of two variables.

## Types of function

Functions are primarily classified as algebraic or transcendental. The former include only those functions which may be expressed in a finite number of terms, involving only the elementary algebraic operations of addition, subtraction, multiplication, division, and root extraction.

Functions are also distinguished as **continuous** or **discontinuous**.
Any function is said to be continuous when an infinitely small change in
the value of the independent variable produces only an infinitely small
change in the dependent variable; and to be discontinuous when an infinitely
small change in the independent variable makes a change in the dependent
variable either finite or infinitely great. All purely algebraic expressions
are continuous functions, as are also such transcendental functions as e^{x},
log *x*, and sin *x*.

**Harmonic** and **periodic** functions are those whose values fluctuate
regularly between certain assigned limits, passing through all of their
possible values, while the independent variable changes by a certain amount
known as the period. Such functions are of great importance in many branches
of mathematical physics. Their essential feature is that, if *f*(*x*)
be a periodic function whose period is *a*, then *f*(*x* + ½*a*) = f(*x* - ½*a*), for all values of *x*.

## The modern view of functions

In modern mathematics, the insistence on specifying an explicit effective
rule has been abandoned; all that is required is that a function *f* associate with every element of some set *X* a unique element of some set *Y*. This makes it possible to prove the
existence of a function without necessarily being able to calculate its
values explicitly. Also, it enables general properties of functions to be
proved independently of their form. The set *X* of all admissible arguments
is called the **domain** of *f*; the set *Y* of all
admissible values is called the **codomain** of *f*. We
write *f *: *X* →*Y*.