Relation: A relation is any set of ordered pairs. Ex: {(-2,3) (3,-8) (5,7) (5,10)} Function: A function is a relation where each x-value (input) corresponds to exactly one y-value (output). Ex: {(-2,3) (3,-8) (5,7) (8,10)}

Function? The button on the machine is the “first variable” or input (x). The bottle that comes out is the output (y). We can have two different buttons (x) that give the same bottle (y). We can NOT have the same button (x) give us two different bottles (y). That would mean the machine is NOT functioning properly.

{(2, 4) (3, 5) (3, 6) (7, 9)} 2) {(2, 4) (4, 5) (6, 8) (7, 9)} Examples {(2, 4) (3, 5) (3, 6) (7, 9)} 2) {(2, 4) (4, 5) (6, 8) (7, 9)} 3) {(1, 2) (3, 5) (7, 6) (8, 9)} 4) {(2, 4) (8, 5) (8, 6) (11, 9)} 5) {(0, 1) (1, 1) (2, 1) (3, 1)} Not a Function Function Function Not a Function Function

Domain: set of all possible values of the first variable (x-values) Range: set of all possible values of the second variable (y-values) Ex: State the domain and range of the relation, and state whether it is a function. { (–7, 5), (4, 12), (8, 23), (16, 8) } domain: { –7, 4, 8, 16} range: { 5, 8, 12, 23 } This is a function because each x-coordinate is paired with only one y-coordinate.

Relations vs. Functions (Tables) x y -4 5 -2 6 2 1 7 3 -5 x y -1 5 6 2 3 7 4 -5 x y -2 5 -1 6 2 3 7 15 FUNCTION Not a FUNCTION FUNCTION

Example State whether the data in each table represents y as a function of x. Explain. x y 2 4 3 6 8 5 x y 3 4 5 -4 6 function not a function

Vertical Line Test What type of lines would pass through the x-axis every time? (Horizontal or Vertical) The vertical line test says: if you can draw a vertical line through more than one point on a graph, then it is NOT a function.

Vertical-Line Test If every vertical line intersects a given graph at no more than one point, then the graph represents a function. function not a function

A Function or Not A Function, That Is The Question!!

Function Notation If there is a correspondence between values of the domain, x, and values of the range, y, that is a function, then y = f(x), and (x,y) can be written as (x,f(x)). The variable x is called the independent variable. The variable y, or f(x) is called the dependent variable.

Example Evaluate f(x) = –2.5x + 11, where x = –1.

Practice Find the indicated outputs. 1) f(x) = x + 3; find f(5), f(-8), and f(-2). 2) g(x) = 3x – x2; find g(0), g(-2), and g(1). 3) p(x) = 2x2 + x - 1; find p(0), p(-2), and p(3).

Example A gift shop sells a specialty fruit and nut mix at a cost of $2.99 per pound. During the holiday season, you can buy as much of the mix as you like and have it packaged in a decorative tin that costs $4.95. a) Write a linear function to model the total cost in dollars, c, of the tin containing the fruit and nut mix as a function of the number of pounds of the mix, n. c(n) = 4.95 + 2.99n b) Find the total cost of a tin that contains 1.5 pounds of the mix. c(n) = 4.95 + 2.99n c(1.5) = 4.95 + 2.99(1.5) c(1.5) = 9.44 $9.44

More Examples Consider the following relation: Is this a function? What is domain and range?

Visualizing domain of

Visualizing range of

Domain = [0, ∞) Range = [0, ∞)