**Basics of Functions**

**Click here for definitions of terms**

A *function* is a mathematical rule that associates every element in one set – called the *domain* of the function – with a unique element in some other (or possibly the same) set, called the *range*. For example, if the function named f is defined by the polynomial 3x + 1, then both the domain and the range of the function f is the set of real numbers. Thus, if x is any real number, then the function f associates x with the value 3x + 1. This is written as f(x) = 3x + 1. This is said “f of x = 3x + 1.” Different values for x and f(x) are shown in the following table:

x | f(x) = 3x + 1 |

0 | 1 |

2 | 7 |

3 | 10 |

-1 | -2 |

-2 | -5 |

Thus, if x is 0, then f(x) is 1; if x is 2, f(x) is 7, etc. We can shorten this by saying, f(0) = 1, f(2) = 7, etc. Notice that we replace the x in the expression f(x) and in the polynomial 3x + 1 with the number 0 or the number 2, etc., so, for example, f(2) = 3·2 + 1 = 6 + 1 = 7.

**Examples of Functions**

*Absolute value *is a function which for any real number x gives the distance x is from 0. If x is a positive real number, then its absolute value, |x|, is x. For example, if x is 3, then |x| = |3| = 3. If x is a negative real number, then |x| = -x. For example, if x is -3, then |x| = |-3| = -(-3) = 3. Notice that both 3 and -3 are the same distance, 3, from 0.

A *sequence* is a function whose domain is usually the set of non-negative integers. Its range can vary, but is usually some subset of the *real numbers*. It is often shown as a series of numbers with the property that each number after the first has a particular relationship to its immediate predecessor.

Example: In this sequence, 4, 7, 10, 13, … each number is 3 greater than the number preceding it. This is an example of an arithmetic sequence. The value of each number in a sequence can be determined if we know the value of the first number and the rule that determines for any given number what the next number will be.

*Statistical functions* return a number which is calculated from a set of values. The particular function is intended to identify something about the set as a whole. Commonly used statistical functions include the mean, median, mode, range of a set of values, variance, and standard deviation.