Recognizing a Function LAUNCH (7 MIN) Before Why would someone hire a dog walking service? During Do you know exactly what it would cost to hire Friendly Dog Walking? After How does each service encourage its clients to sign up for a greater number of hours? PART 1 (6 MIN) During the Intro In the ordered pair 3, 2, what are the input and output? What does the arrow represent in a mapping diagram? What is one way you could find the correct set for a mapping diagram? KEY CONCEPT (3 MIN) Note to students that a function can have two arrows pointing to the same output. Andie Says (Screen 1) Use the Andie Says button to help students see that a mapping diagram is an excellent visual tool you can use to identify functions. PART 2 (6 MIN) While solving the problem What do you need to do to draw a mapping diagram? In a mapping diagram, what shows you that the relation is a function? While solving part (b) Why don t the values in the input or output repeat? PART 3 (5 MIN) Andie Says (Screen 1) Use the Andie Says button to explore how to use a table to identify a relationship between two variables. How can you identify whether a relation in a table is a function? After solving the problem Which do you find easier to identify a function a table, a set of ordered pairs, or a mapping diagram? PART 4 (5 MIN) Does it matter whether you start on the left side of a graph and move right or start on the right side of the graph and move left? While solving part (b) In some cases, the pencil does not pass through any points. What does that tell you? CLOSE AND CHECK (8 MIN) Describe how you identify a function using a mapping diagram. Describe how you identify a function using a graph.
Recognizing a Function LESSON OBJECTIVES 1. Define a function as a rule that assigns to each input exactly one output. 2. Define a function as a table of values that assigns to each input exactly one output. FOCUS QUESTION How can you know a function when you see one? MATH BACKGROUND Students have studied the coordinate plane and graphing ordered pairs in the coordinate plane. They also graphed lines in the topic Proportional Relationships, Lines, and Linear Equations. Students will use these skills as they are introduced to and explore functions. Functions are presented as a special type of relation in which each x-value, or input, corresponds to exactly one y-value, or output. Relations and functions are represented in this topic as a set of ordered pairs, as a mapping diagram, a table, a graph, and/or an equation. Students learn to move between these different representations and identify relations that are functions by verifying that each input has only one output. At the end of this lesson, students use the vertical line test to determine whether relations, presented as graphs, are also functions. Notice that most methods for identifying functions involve looking for characteristics that indicate the relation is not a function. If the method does not uncover any such problems, then the relation is a function. In this lesson and the entire topic, function notation, such as f x, and the terms domain and range are purposely not used. Students do not even use function rules involving variables until the next lesson. You may want to share that relations and functions do not need to be a set of ordered pairs. In the next lesson, students will extend their work with multiple representations of functions to include verbal descriptions and function rules (two-variable equations). In future lessons, they will learn to distinguish between linear and nonlinear functions and increasing, decreasing, and constant intervals, and to match graphs with written function rules. LAUNCH (7 MIN) Objective: Recognize a situation that describes a function. Students find the fees associated with two different dog walking services to find one quantity given another. This problem prepares them to find output values for given input values. They also recognize the usefulness of a known output over a varied cost. Before Why would someone hire a dog walking service? [Sample answer: They work or travel and need someone to walk their dog.] During Do you know exactly what it would cost to hire Friendly Dog Walking? [Sample answer: No, but you have a range because the amount varies based on the flip of a coin.]
After How does each dog walking service encourage its clients to sign up for a greater number of hours? [Sample answer: Both services charge less per hour if you hire them for a greater number of hours. For instance, Friendly Dog Walking charges $6 8 for 3 hours, which is $2/hr or $2.66/hr. They charge $2 4 for 1 hour.] Students can calculate the total fees however they like as long as they realize that they need to purchase one of each type of service for the weekend. They may choose Dog Walk Friends because the price is known and they are not interested in taking a risk. Others may choose Friendly Dog Walking because they like the risk and because there is potential to pay less than they would pay Dog Walk Friends. Connect Your Learning Move to the Connect Your Learning screen. Use the Launch to discuss how a quantity can depend on another quantity. In this case, the cost depends on the number of hours. Students can identify the relationship between two variables in a real-world situation; they first saw a dependent relationship in Grade 6, when they looked at two-variable functions. PART 1 (6 MIN) Objective: Use a mapping diagram to link input values to output values. Students learn that relations can be expressed as ordered pairs and in mapping diagrams. They match sets of ordered pairs to their corresponding mapping diagrams. This problem helps students realize that there is more than one way to represent a function. Instructional Design Use the Intro to present the definition of relation, input, output, and mapping diagrams. Move to Screen 2. Have students drag each set of ordered pairs to the correct mapping diagram and justify their reasoning. When all sets of ordered pairs have been placed, click the Check button. Any incorrect answers will snap back to their original locations. Give students an opportunity to connect the coordinates to the numbers in the mapping diagram and correct their own errors. During the Intro In the ordered pair 3, 2, what are the input and output? [Sample answer: The x-coordinate, 3, is the input, and the y-coordinate, 2, is the output.] What does the arrow represent in a mapping diagram? [The arrow shows which output values are associated with a given input. So, the arrowheads always point from left to right.] What is one way you could find the correct set for a mapping diagram? [Sample answer: Examine the input and output values and write the ordered pairs they represent. Each arrow represents one ordered pair.]
After solving the problem Which mapping diagram is probably easiest to match with a set of ordered pairs? [Sample answer: The one with just two inputs of 1 and 4, because I only had to look for the x-values of 1 and 4.] Students may prefer to examine a set of ordered pairs and then look at each mapping diagram to see if it matches the ordered pairs. As needed, clarify that 0, 1 and 1, 0 are not the same and appear different in the mapping diagram. Students can write a set of ordered pairs as a mapping diagram, or they can show how a mapping diagram represents a set of ordered pairs. Welcome students to show both methods as you solve the problem. Differentiated Instruction For struggling students: Clarify the terms input and output to help students remember the order within each ordered pair. Tell students that input is the value you substitute into the function, and output is the value you get out of the function. For advanced students: Have students create their own mapping diagrams in which the input is the year (between 1789 and the present) and the output is the president. Students might have to research which president was in office for certain years. Error Prevention Some students get mixed up about how ordered pairs relate to mapping diagrams. Point out that just like x comes before y alphabetically, input is before output. Furthermore, x is the first coordinate and the input is the first group in a mapping diagram. Therefore, x corresponds to input, and y corresponds to output. Got It Notes If you show answer choices, consider the following possible student errors: Students who select A probably switched the x- and y-coordinates, or input and output values. Students who choose C may not understand what the arrows indicate because they formed ordered pairs by grouping numbers within each column. If students choose D, they may not realize that the order of the coordinates matters. KEY CONCEPT (3 MIN) Teaching Tips for the Key Concept A function is a special kind of relation. All functions are relations, but not all relations are functions. In a mapping diagram, functions have exactly one arrow directed away from each input value. So, if a single input corresponds to two or more outputs, the relation is not a function. However, a function can have two arrows pointing to the same output. Consider returning to the previous Got It to examine such a function, because the Key Concept does not present this possibility. Students may not initially understand that functions are a subset of relations. One interesting example is texting on a cell phone keypad. Each letter in the alphabet corresponds to exactly one number, which is a function. When you send a text message, however, you are inputting a number, which outputs a letter. Since each number corresponds to more than one letter, texting is not a function but is still a relation.
Andie Says (Screen 1) Use the Andie Says button to help students see that a mapping diagram is an excellent visual tool you can use to identify functions. What should you look for when deciding whether a relation is a function? [Sample answer: Look for two arrows whose tails start from the same input. If that happens, the relation is not a function.] PART 2 (6 MIN) Objective: Identify a function, given a set of ordered pairs, using a mapping diagram. Students transition from interpreting mapping diagrams to making their own and identifying whether each relation is a function. Mapping diagrams illustrate the definition of function because each input can have only one output, meaning only one arrow can be directed away from each input. Instructional Design Have students create the mapping diagrams on the whiteboard. Let one student write all the distinct inputs, another all the distinct outputs, and a third draw the arrows. What is the definition of a function? [Sample answer: Each input value produces exactly one output value.] While solving the problem What do you need to do to draw a mapping diagram? [Sample answer: Write the input values in a column and the output values is a second column, and label each column accordingly. Then draw arrows directed from each input value to its corresponding output value to represent each ordered pair.] In a mapping diagram, what shows you that the relation is a function? [Sample answer: Each input has only one arrow directed away from it. While solving part (b) Why don t the values in the input or output repeat? [Sample answer: You only need to write each input/output once because you can have it connect to multiple arrows if needed.] Students may quickly identify relations that are not functions by finding repeated x-values. However, encourage students to make the mapping diagrams because it can help reinforce that the x-values are the input values and y-values are the output values. There is no standard order to the values in either the input or the output part of a mapping diagram, but each value should appear only once. Differentiated Instruction For struggling students: To help students understand why part (b) is a function but part (c) is not, do an activity involving students heights. Let each student be an input and their height be the output. Two students can have the same height, but no student can have more than one height. So two inputs can lead to the same output, but no input can have two outputs (heights).
For advanced students: Have students draw one mapping that was a function (part (a) or part (b)) and one mapping that was not (part (c)) on the coordinate plane. See if they can notice a difference between the two. This prepares students to use the vertical line test to determine whether a relation is a function. Got It Notes Students should quickly recognize that the set of ordered pairs repeats the same x-value of 1, so it cannot be a function. Students may not need a mapping diagram to solve this problem, but make sure they explain their reasoning. PART 3 (5 MIN) Objective: Identify a function, given a table of values, using a mapping diagram. Students recall that ordered pairs can be written as rows of a table and therefore define relations using tables. They determine whether each relation is a function in a way similar to those involving mapping diagrams. Instructional Design Have students circle any repeated input and output values. Discuss which column of the table will contain a repeated value if the relation is not a function. Then have students create the corresponding mapping diagrams. During the Intro What does the ordered pair 1, 12 represent in the table? [1 hour worked for pay of $12.] Andie Says (Screen 1) Use the Andie Says button to explore how to use a table to identify a relationship between two variables. How is a table similar to a mapping diagram? [Sample answer: Inputs are on the left, and outputs are on the right. Each input is associated with an output by a row in the table instead of an arrow.] How can you identify whether a relation in a table is a function? [Sample answer: Look for repeated input values. If you find one or more repeated input values (with different output values), the relation is not a function.] After solving the problem Which do you find easier to identify a function a table, a set of ordered pairs, or a mapping diagram? [Sample answers: A table; I can quickly see if two inputs are repeated. A mapping diagram; I can look for two arrows starting from the same input.] Students may be able to identify relations that are not functions by looking for repeated input values. Others may need to make the corresponding mapping diagram to see how the information in the table tells them whether the relation is a function. Still other students may want to write the ordered pairs associated with each table. Showing multiple representations helps students remember how the input and output values are related.
Got It Notes Use this problem to show students that a function does not depend on what the input and output values are but on the relationship between these values. This can be visually observed in a mapping diagram by covering up the output values and how students could still say that the relation is a function because of the single lines directed away from each input value. PART 4 (5 MIN) Objective: Identify a function, given a graph, using the vertical line test. The final model that students use to represent relations in this lesson is a graph. They learn the vertical line test, which visually identifies any input that has more than one output. If the pencil passes through at most one point for any input, then the relation is a function. Instructional Design Use the Intro animation to present how to test whether relations represented as graphs are functions. In the Example, have students rotate the pencil and drag it horizontally across each graph to test if the graphed relation is a function. During the Intro In what ways can you represent a relation or function? [Sample answer: graph, table, mapping diagram, set of ordered pairs] Why does the vertical line test work? [Sample answer: Since x is the input, any value on the x-axis with more than one point corresponds to two points with the same x-coordinate and different y-coordinates.] Does it matter whether you start on the left side of a graph and move right or start on the right side of the graph and move left? [It does not matter which side of the graph you start on.] While solving part (b) In some cases, the pencil does not pass through any points. What does that tell you about the relation? [Sample answer: Having no outputs for an input means the same thing as having one output. It says that the relation is still a function until you find more than one output for the same input.] Could you use the vertical line test to see if a set of ordered pairs is a function? [Sample answer: Yes; first you need to graph the ordered pairs.] Some students might just examine each graph carefully to look for any x-values that have repeated y-values. Point out that this is essentially what the vertical line test does. Differentiated Instruction For struggling students: Have students draw a graph that is a horizontal line. Ask them if this graph satisfies the vertical line test and is a function. Then have students describe a real-world situation that can be represented by this function, and represent it in a set of ordered pairs, a table, and a mapping diagram.
For advanced students: Have students describe a real-world situation that can be represented by this function, and represent it in a set of ordered pairs, a table, and a mapping diagram. Got It Notes Teach students an important test-taking strategy involving the process of elimination. If they determine graph I is not a function, they can eliminate choices A and D. Therefore, they know that only one of graphs II and III can be a function. Got It 2 Notes Use mapping diagrams to make the reasoning clear. A horizontal line is like having infinitely many inputs all pointing to the same output. A vertical line is like having infinitely many outputs all coming from the same input, which is not a function. CLOSE AND CHECK (8 MIN) Focus Question Sample Answer By checking the relationship between the two related quantities for specific conditions, you can determine whether a given relation is a function. Focus Question Notes Students may describe how you can use a mapping diagram, table, or graph to identify whether a relationship is a function. Essential Question Connection This lesson addresses the Essential Question "What is a function?" as students learn to identify a function using a mapping diagram, table, or graph. Describe how you identify a function using a mapping diagram. [Sample answer: A function requires that each input value produce exactly one output value. So if a mapping diagram shows inputs and outputs, each input can have only one arrow starting from it.] Describe how you identify a function using a graph. [Sample answer: You can use a pencil to perform the vertical line test. You hold the pencil vertically and move it slowly across the graph from left to right. If at any time the pencil crosses the graph more than once, the graph shows a relationship that is not a function.]