Functions and Equations | Research & Encyclopedia Articles

In mathematics, function is a central idea. Imagine a machine that takes numbered balls from 1 through 26 and labels them with the English alphabet letters A through Z. This machine mimics a mathematical function. A function takes an object from one set A (the input) and maps it to an object in another set B (the output). In mathematics, A and B are usually sets of numbers. In symbols, this relationship is written as f: A → B.

So, a function f is the name of a relationship between two sets. Functions are usually denoted by the letters f, g, or h. A is called the domain (input), and B is called the range (output). If the elements of the domain are denoted by x, and the elements of the range are denoted by y, then a function can also be written as y = f(x). This is read as "y is a function of x." Notice that this notation does not mean that f is multiplied by x. Instead, the value of f depends on the value of x.

A simple example of a function is y = f(x), where f(x) = x + 2. To each number x, add 2 to get y. When x is 3, y is 5, and when x is 4, y is 6. The value y of the function, f(x), depends on the choice of x. The input, or x, is called the independent variable, and the output, or y, is called the dependent variable.

Another example is a relationship between the positive integer set (domain) and the even number set (range). To each positive integer n, the function f(n) assigns a value of 2n. In symbols, f(n) = 2n.

In a function, each element of the domain must map to exactly one element of the range. However the opposite is not true. For example, f(x) = |x| is a function. Each value of f(x) corresponds to two values of x.

Now consider a function g with the real number set as the domain set. To each number x, g assigns 3 times x. That is, g(x) = 3x.

Functions are visualized geometrically by plotting their graphs on a Cartesian plane. You can plot a function by taking a few numbers from the domain sets and finding their functional values. For example, g(x) = 3x would yield the points (-1, 3), (0, 0), and (1, 3). These points can be connected by a straight line.

In functions such as f(x) = 3x, g(x) = x + 2, or h(x) = (½)x, the power of the independent variable, x, is 1. Such functions are called linear functions. Plotting the graph of linear functions always produces straight lines. In contrast, consider the function f(x) = x²; its graph is not a straight line but rather a parabola.

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Miller, Charles D., Vern E. Heeren, and E. John Hornsby, Jr. Mathematical Ideas, 9th ed. Boston: Addison-Wesley, 2001.

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