Relations and Functions

Def: Relation A relation is a set of ordered pairs. The domain is the set of all abscisses (x-values) and the range is the set of ordinates (y-values). Ex 1 If x is a positive integer less than 6 and y = 5 + x, list the ordered pairs that satisfy the relation. State the domain and range of the relation.

Def: Function A function is a relation in which each element of the domain is paired with exactly one element of the range. Ex 2 Which relations below are functions? a. Names and social security numbers b. Addresses and names. c. {(2,4) (-2, 5) (3, 7)} d. {(4,1) (4, 3) (5, 6)} e. { (2,5) (3, 5) (4,5)}

Ex 2 Are the relations below functions? a. b.

Vertical line test. If you draw a vertical line on the graph of the relation and the vertical lines crosses 2 different points then the relation is not a function. Ex 3 Which of the graphs at the right represent functions? a b

Function notation  f(x) reads “f of x”  Interpreted as the value of function f at x.  x is the independent value and f(x) is the dependent value. y = 2x + 1 can also be written as f(x) = 2x + 1. (x, y) is the same as (x, f(x))

Ex 4 a. If f(x) = 4x 2 - 2x + 5 Find f(4). b) Find f(-3).

Greatest Integer Function f(x) = [x] means the greatest integer less than or equal to x. Ex 5 f(x) = [x] Find f(5); f(5.1); f(4.1); f(4.6) and f(4.9) Graph f(x) = [x]

Once an equation of a function is given but the domain is not specified, the domain consists of all real numbers for which the corresponding values in the range are also real numbers. (The domain cannot contain values for which the range is undefined.) Ex 6 Give the domain for the following functions. a. b. c. d.