Functions and Equations

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Functions and Equations

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.

Examples of Functions

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.

Function Notation and Graphs

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 ) = x ²; its graph is not a straight line but rather a parabola .