Name Class Date. Understanding Functions
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1 Name Class Date 3-2 Relations and Functions Going Deeper Essential question: How do you represent functions? F-IF.. ENGAGE Understanding Functions A set is a collection of items called elements. A function pairs each element in one set, called the domain, with exactly one element in a second set, called the range. For example, the function below pairs each element in the domain with its square. Domain Range A function can be described using function notation. The function f assigns the output value f (x) in the range to the corresponding input value x from the domain. The notation f (x) is read as f of x. (It does not indicate the product f times x. ) For the function shown above, f (3) = 9. REFLECT a. The domain of the function can be written using set notation as {0,, 2, 3, 4}. Write the range of the function using set notation. b. Tell how to read the statement f (4) = 6. Then interpret what it means in terms of input and output values. c. Suppose the 3 were paired with the 4 instead of the 9. Would the pairing of the two sets still be a function? Why or why not? d. Suppose the 3 were paired with the 4 and the 9. Would the pairing of the two sets still be a function? Why or why not? e. If you pair each month with all the possible numbers of days in the month, will you get a function? Why or why not? Chapter 3 2 Lesson 2
2 Functions are often used to describe a relationship between two variables. The independent variable represents an input value of the function and the dependent variable represents an output value. An algebraic expression that defines a function is a function rule. For example, x 2 is the function rule for the squaring function f (x) = x 2. If you know a value for the independent variable, you can use a function rule to find the corresponding value for the dependent variable. 2 F-IF..2 EXAMPLE Representing Discrete Linear Functions The cost of sending m text messages at $0.25 per message can be represented by the function C(m) = 0.25m. A Complete the table for the given domain values. Write the results as ordered pairs in the form (independent variable, dependent variable). Independent variable, m Dependent variable, C(m) = 0.25m (m, C(m)) (0) = 0 (0, 0) B Choose a beginning, an end, and a scale for the vertical axis. C Graph the function by plotting the ordered pairs. The independent variable goes on the horizontal axis and the dependent variable on the vertical axis. Use scales that will make it easy to read points. Label the graph. C(m) m Chapter 3 22 Lesson 2
3 REFLECT 2a. Use function notation to represent the cost of sending 5 text messages. Evaluate the function for that value. Include units. 2b. Suppose that the domain of the function is not limited as in the Example. Describe a reasonable domain of the function. 2c. Suppose that the domain of the function is not limited as in the Example. Describe a reasonable range of the function. 2d. Is the independent variable represented by the horizontal axis or the vertical axis? Why does this make sense? 2e. Would it make sense to connect the points on the graph with a line? Why or why not? 2f. The figure below shows a representation of the function rule. Explain what is being shown in the context of the situation. Input Rule: C(m) = 0.25m Output g. Suppose the cost per text message were $0.20 instead of $0.25. Then the cost of sending m text messages could be represented by the function C(m) = 0.2m. Describe a reasonable domain and range for this function. Chapter 3 23 Lesson 2
4 3 F-IF.2.5 EXAMPLE Representing Discrete Nonlinear Functions Ben wants to tile part of a floor with 36 square tiles. The tiles come in whole-number side lengths from 2 to 6 inches. If s is the side length of a tile, the area that he can cover is A(s) = 36 s 2. A Identify the domain of the function. B Make a table of values for this domain. Write the results as ordered pairs in the form (independent variable, dependent variable). Independent variable, s Dependent variable, A(s) = 36 s 2 (s, A(s)) C Choose a beginning, an end, and a scale for the vertical axis. D Graph the function by plotting the ordered pairs. A(s) s Chapter 3 24 Lesson 2
5 REFLECT 3a. Identify the range of the function. 3b. What does A(3) = 324 mean in this context? 3c. Describe another reasonable beginning, end, and scale for the vertical axis. Include units. 3d. Do the points appear to lie in a straight line? PRACTICE Tell whether each pairing of numbers describes a function. If so, identify the domain and the range. If not, explain why not.. Each whole number from 0 to 9 is paired with its opposite. 2. Each odd number from 3 to 9 is paired with the next greater whole number. 3. The whole numbers from 0 to 2 are paired with their factors. 4. Each even number from 2 to 0 is paired with half the number. 5. {(36, 6), (49, 7), (64, 8), (8, 9), (36, -6), (49, -7), (64, -8), (8, -9)} 6. {(-64, -4), (-27, -3), (-8, -2), (-, -), (0, 0), (, ), (8, 2), (27, 3), (64, 4)} Chapter 3 25 Lesson 2
6 7. Whitley has a $5 gift card for music downloads. Each song costs $ to download. The amount of money left on the card can be represented by the function M(d) = 5 - d, where d is the number of songs she has downloaded. a. Make a table and graph the function. d M(d) (d, M(d)) M(d) d b. Identify the domain and range of the function and the units of the independent and dependent variables. 8. Ben wants to cover a table that has an area of 864 square inches. The function T(s) = gives the number of tiles he needs with side length s. The tiles come in s side lengths of in., 4 in., 6 in., and 2 in. a. Make a table and graph the function. s T(s) (s, T(s)) T(s) s b. Identify the domain and range of the function and the units of the independent and dependent variables. Chapter 3 26 Lesson 2
7 Name Class Date 3-2 Additional Practice Express each relation as a table, as a graph, and as a mapping diagram.. {( 5, 3), ( 2, ), (, ), (4, 3)} x y 2. {(4, 0) (4, ), (4, 2), (4, 3), (4, 4), (4, 5)} x y Give the domain and range of each relation. Tell whether the relation is a function. Explain x y D: D: D: R: R: R: Function? Function? Function? Explain: Explain: Explain: Chapter 3 27 Lesson 2
8 Problem Solving Give the domain and range of each relation and tell whether it is a function.. The mapping diagram shows the ages 2. and grade level of four children. Age Shoe Size The list represents the number of cars sold and the bonus received by the salespeople of a car dealership. {(, 50), (2, 50), (3, 00), (4, 50)} 4. A 2-inch-tall plant grows at a rate of 2.5 inches every week for 5 weeks. Let represent the number of weeks and represent the height of the plant. Use the graph below to answer questions 5 6. A conservation group has been working to increase the population of a herd of Asian elephants. The graph shows the results of their efforts. Select the correct answer. 5. Which relation represents the information in the graph? A {(, 4.5), (2, 6), (3, 0), (4, 4.5)} B {(, 5), (2, 6), (3, 0), (4, 5)} C {(4.5, ), (6, 2), (0, 3), (4.5, 4)} D {(5, ), (6, 2), (0, 3), (5, 4)} 6. What is the range of the relation shown in the graph? F {0,, 2, 3, 4, 5} G {, 2, 3, 4} H {4.5, 6, 0, 4.5} J {5, 6, 0, 5} Chapter 3 28 Lesson 2
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