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1. 1-6 Relations and Functions Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

2. Warm Up Use the graph for Problems 1–2. 1. List the x-coordinates of the points. 2. List the y-coordinates of the points. –2, 0, 3, 5 3, 4, 1, 0

3. Objectives Identify the domain and range of relations and functions. Determine whether a relation is a function.

4. Vocabulary relation domain range function

5. A relation is a pairing of input values with output values. It can be shown as a set of ordered pairs (x,y), where x is an input and y is an output. The set of input values for a relation is called the domain, and the set of output values is called the range.

6. Mapping Diagram Domain Range A 2 B C Set of Ordered Pairs {(2, A), (2, B), (2, C)} (x, y) (input, output) (domain, range)

7. Example 1: Identifying Domain and Range Give the domain and range for this relation: {(100,5),(120,5), (140,6), (160,6), (180,12)}. List the set of ordered pairs: {(100, 5),(120, 5), (140, 6), (160, 6), (180, 12)} Domain: {100, 120, 140, 160, 180} The set of x-coordinates. Range: {5, 6, 12} The set of y-coordinates.

8. Check It Out! Example 1 Give the domain and range for the relation shown in the graph. List the set of ordered pairs: {(–2, 2), (–1, 1), (0, 0), (1, –1), (2, –2), (3, –3)} Domain: {–2, –1, 0, 1, 2, 3} The set of x-coordinates. Range: {–3, –2, –1, 0, 1, 2} The set of y-coordinates.

9. Suppose you are told that a person entered a word into a text message using the numbers 6, 2, 8, and 4 on a cell phone. It would be difficult to determine the word without seeing it because each number can be used to enter three different letters.

10. Number {Number, Letter} The numbers 6, 2, 8, and 4 each appear as the first coordinate of three different ordered pairs. {(6, M), (6, N), (6, O)} {(2, A), (2, B), (2, C)} {(8, T), (8, U), (8, V)} {(4, G), (4, H), (4, I)}

11. However, if you are told to enter the word MATH into a text message, you can easily determine that you use the numbers 6, 2, 8, and 4, because each letter appears on only one numbered key. The first coordinate is different in each ordered pair. {(M, 6), (A, 2), (T, 8), (H,4)} A relation in which the first coordinate is never repeated is called a function. In a function, there is only one output for each input, so each element of the domain is mapped to exactly one element in the range.

12. Although a single input in a function cannot be mapped to more than one output, two or more different inputs can be mapped to the same output.

13. Not a function: The relationship from number to letter is not a function because the domain value 2 is mapped to the range values A, B, and C. Function: The relationship from letter to number is a function because each letter in the domain is mapped to only one number in the range.

14. Example 2: Determining Whether a Relation is a Function Determine whether each relation is a function. A. from the items in a store to their prices on a certain date There is only one price for each different item on a certain date. The relation from items to price makes it a function. B. from types of fruits to their colors A fruit, such as an apple, from the domain would be associated with more than one color, such as red and green. The relation from types of fruits to their colors is not a function.

15. Check It Out! Example 2 Determine whether each relation is a function. A. There is only one price for each shoe size. The relation from shoe sizes to price makes is a function. B. from the number of items in a grocery cart to the total cost of the items in the cart The number items in a grocery cart would be associated with many different total costs of the items in the cart. The relation of the number of items in a grocery cart to the total cost of the items is not a function.

16. Every point on a vertical line has the same x-coordinate, so a vertical line cannot represent a function. If a vertical line passes through more than one point on the graph of a relation, the relation must have more than one point with the same x-coordinate. Therefore the relation is not a function.

18. Example 3A: Using the Vertical-Line Test Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is a function. Any vertical line would pass through only one point on the graph.

19. Example 3B: Using the Vertical-Line Test Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is not a function. A vertical line at x = 1 would pass through (1, 1) and (1, –2).

20. Check It Out! Example 3a Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is a function. Any vertical line would pass through only one point on the graph.

21. Check It Out! Example 3a Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. This is not a function. A vertical line at x = 1 would pass through (1, 2) and (1, –2).

22. Lesson Quiz: Part I 1. Give the domain and range for this relation: {(10, 5), (20, 5), (30, 5), (60, 100), (90, 100)}. Determine whether each relation is a function. 2. from each person in class to the number of pets he or she has 3. from city to zip code D: {10, 20, 30, 60, 90)} R: {5, 100} function not a function

23. Lesson Quiz: Part II Use the vertical-line test to determine whether the relation is a function. If not, identify two points a vertical line would pass through. 4. not a function; possible answer: (3, 2) and (3, –2)

24. 1-7 Function Notation Warm Up Lesson Presentation Lesson Quiz Holt Algebra 2

25. Warm Up Evaluate. 1. 5x – 2 when x = 4 18 2. 3x2 + 4x – 1 when x = 5 94 3. when x = 16 48 4. 2 – t2 when 5. Give the domain and range for this relation: {(1, 1), (–1, 1), (2, 4), (–2, 4), (–3, 9), (3, 9)}. D: {–3, –2, –1, 1, 2, 3} R: {1, 4, 9}

26. Objectives Write functions using function notation. Evaluate and graph functions.

27. Vocabulary function notation dependent variable independent variable

28. Some sets of ordered pairs can be described by using an equation. When the set of ordered pairs described by an equation satisfies the definition of a function, the equation can be written in function notation.

29. Output value Input value Output value Input value ƒ(x) =5x+ 3 ƒ(1) =5(1)+ 3 ƒ of 1 equals 5 times 1 plus 3. ƒ of x equals 5 times x plus 3.

30. The function described by ƒ(x) = 5x + 3 is the same as the function described by y = 5x + 3. And both of these functions are the same as the set of ordered pairs (x, 5x+ 3). y = 5x + 3 (x, y) (x, 5x + 3) Notice that y = ƒ(x) for each x. ƒ(x) = 5x + 3 (x, ƒ(x)) (x, 5x + 3) The graph of a function is a picture of the function’s ordered pairs.

31. Caution f(x) is not “f times x” or “f multiplied by x.” f(x) means “the value of f at x.” So f(1) represents the value of f at x =1

32. For each function, evaluate ƒ(0), ƒ , and ƒ(–2). ƒ = 8 + 4 =10 Example 1A: Evaluating Functions ƒ(x) = 8 + 4x Substitute each value for x and evaluate. ƒ(0) = 8 + 4(0) =8 ƒ(–2) = 8 + 4(–2)=0

33. Example 1B: Evaluating Functions For each function, evaluate ƒ(0), ƒ , and ƒ(–2). Use the graph to find the corresponding y-value for each x-value. ƒ(0) = 3 ƒ = 0 ƒ(–2) = 4

34. For each function, evaluate ƒ(0), ƒ , and ƒ(–2). Check It Out! Example 1a ƒ(x) = x2 – 4x

35. For each function, evaluate ƒ(0), ƒ , and ƒ(–2). Check It Out! Example 1b ƒ(x) = –2x + 1

36. In the notation ƒ(x), ƒis the name of the function. The output ƒ(x) of a function is called the dependent variable because it depends on the input value of the function. The input x is called the independent variable. When a function is graphed, the independent variable is graphed on the horizontal axis and the dependent variable is graphed on the vertical axis.

38. Example 2A: Graphing Functions Graph the function. {(0, 4),(1, 5), (2, 6), (3, 7), (4, 8)} Graph the points. Do not connect the points because the values between the given points have not been defined.

39. Reading Math A function whose graph is made up of unconnected points is called a discrete function.

40. Example 2B: Graphing Functions Graph the function f(x)= 3x – 1. Make a table. Graph the points. Connect the points with a line because the function is defined for all real numbers.

41. 3 5 7 9 2 6 10 Check It Out! Example 2a Graph the function. Graph the points. Do not connect the points because the values between the given points have not been defined.

42. Check It Out! Example 2b Graph the function f(x)= 2x + 1. Graph the points. Make a table. Connect the points with a line because the function is defined for all real numbers.

43. The algebraic expression used to define a function is called the function rule. The function described by f(x) = 5x + 3 is defined by the function rule 5x + 3. To write a function rule, first identify the independent and dependent variables.

44. Example 3A: Entertainment Application A carnival charges a $5 entrance fee and $2 per ride. Write a function to represent the total cost after taking a certain number of rides. Letrbe the number of rides and let C be the total cost in dollars. The entrance fee is constant. First, identify the independent and dependent variables. Cost depends on the entrance fee plus the number of rides taken Dependent variable Independent variable Cost = entrance fee + number of rides taken C(r) = 5 + 2r Replace the words with expressions.

45. Example 3B: Entertainment Application A carnival charges a $5 entrance fee and $2 per ride. What is the value of the function for an input of 12, and what does it represent? Substitute 12 for r and simplify. C(12) = 5 + 2(12) C(12) = 29 The value of the function for an input of 12 is 29. This means that it costs $29 to enter the carnival and take 12 rides.

46. Check It Out! Example 3a A local photo shop will develop and print the photos from a disposable camera for $0.27 per print. Write a function to represent the cost of photo processing. Let x be the number of photos and let f be the total cost of the photo processing in dollars. First, identify the independent and dependent variables. Cost depends on the number of photos processed Dependent variable Independent variable Cost = 0.27 number of photos processed f(x) = 0.27x Replace the words with expressions.

47. Check It Out! Example 3b A local photo shop will develop and print the photos from a disposable camera for $0.27 per print. What is the value of the function for an input of 24, and what does it represent? f(24) = 0.27(24) Substitute 24 of x and simplify. = 6.48 The value of the function for an input of 24 is 6.48. This means that it costs $6.48 to develop 24 photos.

48. Lesson Quiz: Part I For each function, evaluate 1.f(x) = 9 – 6x 9; 6; 21 2. 4; 6; 0 3. Graph f(x)= 4x + 2.

49. Lesson Quiz: Part II 4. A painter charges $200 plus $25 per can of paint used. a. Write a function to represent the total charge for a certain number of cans of paint. t(c) = 200 + 25c b. What is the value of the function for an input of 4, and what does it represent? 300; total charge in dollars if 4 cans of paint are used.