Functions (mathematics)

Summary: There are many different types of functions that arose in a variety of historical contexts.

A mathematical function expresses the idea that one quantity can be completely determined by another quantity. As such, a function is a well-defined rule between two sets, A and B, where each element x of A is assigned to one element y of B. The careful study of the implications and applications of this definition comprises much of mathematics. Functions are ubiquitous throughout nearly all fields of mathematics and their importance cannot be overstated. Since a function expresses a relationship between an independent variable and a dependent variable, many real-world phenomena are modeled using functions. Functional correspondence between variables can be expressed verbally, algebraically, or graphically. Although the formal definition of a function is a relatively recent development, this concept has been implicit since the beginning of mathematics.

Examples of Functions

Functions can be classified in various ways and many of these specific classes of functions are the bases of entire fields of study within mathematics. Below are a few examples to illustrate some areas encompassed by the definition of a function. The algebraic equation y = x² expresses y as a function of x. No matter what value has been chosen for x, only one y value will be produced.

The expression y = ∓x does not represent y as a function of x. If x = 1 is entered into the equation, then two y values are obtained, y = -1 and y = 1.

The expression y = cos(x) is a trigonometric function. For any given angle expressed in radian measure, the cosine function uses a rule from trigonometry to produce a real number from -1 to 1.

An example of an exponential function is given by the formula y = 2^x. For instance, when x=2, y = 4. Exponential functions are useful for situations in which the rate of growth of a population is directly proportional to the size of the population, such as modeling the growth of bacteria or the decay of a radioactive sample.

A piecewise function is defined by different formulas for different values of x that are entered into the function. The Heaviside function is an example of a piecewise function: y = 1 for x≥0, and y = 0 for x<0.

If one assigns “on” to 1 and “off” to 0, this function models the behavior of turning on a switch at x = 0 and leaving it on. It is also used in the study of electric circuits to indicate the surge of an electric current.

A barcode scanner at a supermarket can be thought of as a function. After a particular barcode is scanned, only one price will be displayed.

A computer program that obtains the five-digit zip code for an address acts as a function because every address in the United States has only one zip code assigned to it.

Equivalent Formulations

A helpful way to think about a function is as a machine that produces exactly one output y for each input x. This “machine” could simply be a description or a list of the pairings between x and y. Although this may be easier conceptually, in practice it is unfeasible to list all of the pairings when there is a large number (or even an infinite number) of x values. When one considers an infinite number or a large number of x values, it is more advantageous to have a mathematical formula that precisely relates x and y.

Graphs of Functions

Another way to represent a function is by using a graph. One begins with a set of x values from any subset of the real number line and a function. The function then specifies a y value for each x. This results in a collection of pairings (x, y). Each of these pairs denotes a point, which is plotted on the xy-plane in the two-dimensional Cartesian coordinate system. The collection of all points produced by this process is the graph of the function.

By virtue of the definition of a function, every x value on the graph is paired with, at most, one y value. Thus, any vertical line that is drawn will cross the graph of the function at most one time. This is known as the “vertical line test”: a curve in the xy-plane is the graph of a function if—and only if—no vertical line crosses the curve more than once. As a consequence of the vertical line test, given a curve, it is relatively easy to determine if it is the graph of a function. All nonvertical straight lines are graphs of functions. Circles are not graphs of functions.

History

The notion of a function has been implicit throughout the history of mathematics. Addition is the most fundamental arithmetical operation and, although it was not initially formulated as such, it is a function of two variables. The pair of numbers to be added is the input and the resulting sum is the output of the addition function. Ancient cultures, such as the Babylonians, developed extensive tables of mathematical calculations of the reciprocals and square roots of positive whole numbers. These calculations involve specific functions but were not formulated using the function concept.

In the fourteenth century, Nicole Oresme had a rudimentary grasp of the idea that one changing quantity can be dependent upon another. He depicted this relationship graphically using a method he called the “latitude of forms.” This depiction was the first known attempt of the graphical representation of a function. Throughout the Middle Ages, the latitude of forms continued to be studied; however, further development of the function concept was hampered by the absence of a suitable algebraic framework.

The formal study of functions began in the late seventeenth century with the discovery of calculus. Although in 1692, Gottfried Leibniz first introduced the word “function” in association with the tangent problem in calculus, its first definition did not emerge until 1718 with Johann Bernoulli. The primary representation of a function at that time was from curves that were connected to physical problems.

As the eighteenth century unfolded, algebraic equations were increasingly being used to represent two-dimensional curves. As a result, the emphasis and focus of the function concept evolved from a graphical setting to that of algebra. This shift was evident in Leonhard Euler’s 1748 treatise on functions, Introductio in analysin infinitorium. Euler’s definition of a function was that of an “analytic expression” or an algebraic formula that could contain any combination, the five arithmetic operations: exponentials, logarithms, trigonometric ratios, derivatives, and integrals. Euler’s emphasis on the algebraic formulation of functions was evident, as the first volume of the Introductio contains no graphs.

The middle of the eighteenth century saw a development of the function concept when a controversy arose over the solution to the Vibrating String Problem. Given an elastic string with fixed endpoints and deformed into an initial shape, the string was released and began to vibrate. The problem was to determine a function that would describe the shape of the string at any future time. In 1747, Jean Le Rond d’Alembert produced a solution in the form of an algebraic equation. A year later, Euler verified that this solution was correct but he disagreed that it was the most general. He claimed that d’Alembert had neglected several initial shapes of the string that could be drawn freehand and for which there were no algebraic expressions. Euler also pointed out that other initial shapes could be obtained by piecing together simpler curves. This critique led to the acceptance of functions produced from freehand drawing for which there may not be any algebraic formula and piecewise defined functions.

Another solution to the Vibrating String Problem further complicated matters. In 1753, Daniel Bernoulli solved the problem differently than Euler and d’Alembert and arrived at a seeming contradiction: different mathematical expressions defined the same function. The controversy was not resolved at the time; it remained for Joseph Fourier to expand upon Bernoulli’s idea. Fourier’s solution to the Heat Conduction Problem in 1807 resulted in a revolution of the understanding of a function. Fourier demonstrated that a function could be expressed as an infinite series of sine and cosine functions, now known as a Fourier series. These series demonstrated that two different expressions could define the same function. Following the development of Fourier series, the connection between the geometric and algebraic forms of a function was further solidified. Furthermore, ideas from calculus were reexamined in a new light.

In 1837, Lejeune Dirichlet suggested a definition of “function” that was closely related to the modern definition. Dirichlet emphasized that a function provides a relationship between two variables but allowed for freedom in describing the rule that describes how x and y are related. To show how pathological a function can become, Dirichlet introduced the function y=c for x an irrational number, and y=d≠c for x a rational number.

This badly behaved function cannot be sketched and there is no algebraic equation defining it.

Since the nineteenth century, it was a natural evolution to recast Dirichlet’s definition by using set theory. The modern definition for a function now provides a correspondence between two sets, which may or may not be numerical; for example, functions between algebraic structures like groups or geometric objects like surfaces.

Bibliography

Boyer, Carl B. A History of Mathematics. Princeton, NJ: Princeton University Press, 1985.

Kleiner, Israel. “The Evolution of the Function Concept: A Brief Survey.” College Mathematical Journal 20, no. 4 (1989).

Stewart, James. Calculus: Early Transcendental. 6th ed. Belmont, CA: Thomson Brooks/Cole, 2008.