1.4 | Functions


A function \(f\) from a set \(X\) to a set \(Y\) is a relation that assigns to each element \(x\) in the set \(X\) exactly one element \(y\) in the set \(Y\). The set \(X\) is the domain (or set of inputs) of the function \(f\), and the set codomain \(Y\) contains the range (or set of outputs). In this context, the elements of \(X\) are called arguments of f. For each argument \(x\), the corresponding unique \(y\) in the range is called the function value at \(x\) or the image of \(x\) under \(f\). It is written as \(f(x)\). One says that \(f\) associates \(y\) with \(x\) or maps \(x\) to \(y\). This is abbreviated by
$$
y = f(x).
$$
A general function is often denoted by \(f\). If a function is often used, it may be given a special name as, for example, the cosecant function of a real number \(x\) is denoted by \(\csc(x)\). The argument is often denoted by the symbol \(x\), but in other contexts may be denoted differently, as well. For example, in physics, the velocity of some body, depending on the time, is denoted \(v(t)\). The parentheses around the argument may be omitted when there is little chance of confusion, thus: \(sin x\); this is known as prefix notation.
A function can be defined by any mathematical condition relating each argument (input value) to the corresponding output value. If the domain is finite, a function \(f\) may be defined by simply tabulating all the arguments \(x\) and their corresponding function values \(f(x)\). More commonly, a function is defined by a formula, or (more generally) an algorithm — a recipe that tells how to compute the value of \(f(x)\) given any \(x\) in the domain.

There are many other ways of defining functions. Examples include trigonometric, inverse trigonometric, logarithmic, exponential, polynomial, rational, piecewise definitions, induction or recursion, limits, infinite series, and as solutions to integral and differential equations. In advanced mathematics, some functions exist because of an axiom, such as the Axiom of Choice!

The difference quotient of \(f(x)\) is given by
$$
\frac{f(x+h) – f(x)}{h}
$$

E 1.4 Exercises