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Formalizing Relations and Functions Objectives: To determine whether a relation is a function To find domain and range and use function notation Relation: a pairing of numbers in one set with numbers in another set Domain: the set of x-values Range: the set of y-values Problem #1: Identifying Functions Using Mapping Diagram Identify the domain and range of the relation. Represent the relation with a mapping diagram. Is the relation a function? A) {(-2, 0.5), (0, 2.5), (4, 6.5), (5, 2.5)} Problem #1: Identifying Functions Using Mapping Diagram Identify the domain and range of the relation. Represent the relation with a mapping diagram. Is the relation a function? B) {(6,5), (4,3), (6,4), (5,8)} Problem #1 Got It? Identify the domain and range of each relation. Represent the relation with a mapping diagram. Is the relation a function? Vertical Line Test Another method used to determine if a relation is a function. Look at the graph and determine if any vertical line passes through more than one point of the graph, then the relation is not a function. Problem #2: Identifying Functions Using the Vertical Line Test Is the relation a function? Use the Vertical Line Test. A) {(-4, 2), (-3, 1), (0,-2), (-4, -1), (1, 2)} Problem #2: Identifying Functions Using the Vertical Line Test Is the relation a function? Use the Vertical Line Test. B) = − 2 + 3 Problem #2 Got It? Is the relation a function? Use the Vertical Line Test. Function Notation y = -3x + 1 is the same as = −3 + 1 - f(x) replaces y read “f of x” f is the name of the function, not a variable Used to emphasize that the function value f(x) depends on the independent variable x. - Other letters besides f can also be used, such as g and h. Problem #3: Evaluating a Function A) The function w(x) = 250x represents the number of words w(x) you can read in x minutes. How many words can you read in 8 minutes? Problem #3: Evaluating a Function B) The function Y(x) = 1 3 represents the number of yards Y(x) in x feet. How many yards are there in 1 mile? Problem #3 Got It? The function T(x) = 65x represents the number of words T(x) that Rachel can type in x minutes. How many words can she type in 7 minutes? *Homework Textbook Page 271; #1 – 3, 5 – 17 All Continued… Objectives: To determine whether a relation is a function To find domain and range and use function notation Problem #4: Finding the Range of a Function A) Multiple Choice The domain of f(x) = 1.5x + 4 is {1, 2, 3, 4}. What is the range? Problem #4: Finding the Range of a Function B) The domain of g(x) = 4x – 12 is {1, 3, 5, 7}. What is the range? Problem #4 Got It? What is the range of f(x) = 3x – 2 with domain {1, 2, 3, 4}? Problem #5: Identifying a Reasonable Domain and Range A) You have 3 qt. of paint to paint the trim in your house. A quart of paint covers 100 2 . The function () = 100 represents the area A(q), in square feet, that q quarts of paint cover. What domain and range are reasonable for the function? What is the graph of the function? Problem #5: Identifying a Reasonable Domain and Range B) Lorena has 4 rolls of ribbon to make party favors. Each roll can be used to make 30 favors. The function F(r) = 30r represents the number of favors F(r) that can be made with r rolls. What are a reasonable domain and range of the function? What is a graph of the function? Problem #5 Got It? A car can travel 32 miles for each gallon of gasoline. The function d(x) = 32x represents the distance d(x), in miles, that the car can travel with x gallons of gasoline. The car’s fuel tank holds 17 gal. Find a reasonable domain and range for each function. Then graph the function. *Homework 4 – 6 Worksheet