DRILL (4 HOURS) (20 dollars) (River Paddlers) The graph below shows the cost (c), in dollars, to rent a boat for h hours at two boat companies. At what number of hours will the cost to rent a boat be the same at both companies? 2) What would be the cost (in dollars) for the boat rentals when the price is the same for each company? 3) Which company is cheaper for a 6 hour rental? (4 HOURS) (20 dollars) (River Paddlers)

2.1 Relations and Functions Objectives: State the domain and range of a relation, and tell whether it is a function Write a function in function notation and evaluate it

Textbook Definitions Into Our Definitions A Relation is a set of ordered pairs. The set of the first coordinates is called the domain of the relation. The set of the second coordinates is called the range of the relation. A Function is a relation in which each element of the domain is paired with exactly one element in the range.

Definitions 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)}

Definitions Domain: set of all possible values of the first variable (x-values) Range: set of all possible values of the second variable (y-values)

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

Example 1 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

Relations vs. Functions (Mapping)

Example 2 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 3 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 4 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

Homework p.60-61 #’s 17 – 28 #’s 36, 37, 40, 41, 46 – 49, 52, 53