# Functions and Equations

# 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

**(the output). In mathematics,**

*B***and**

*A***are usually sets of numbers. In symbols, this relationship is written as**

*B**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

**is called the**

*B***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 2*n.* In symbols, *f* (*n* ) = 2*n.*

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* ) = 3*x.*

## 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* ) = 3*x* 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* ) = 3*x, 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** .

see also Mapping; Mathematical.

*Rafiq Ladhani*

## Bibliography

Amdahl, Kenn, and Jim Loats. *Algebra Unplugged.* Broomfield, CO: Clearwater Publishing Co., 1995.

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

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**Functions and Equations**