Lesson 10
Domain and Range (Part 1)
10.1: Number of Barks (5 minutes)
Warm-up
This warm-up prompts students to consider possible input and output values for a familiar function in a familiar context. The work here prepares students to do the same in other mathematical contexts and to think about domain and range in the rest of the lesson.
Student Facing
Earlier, you saw a situation where the total number of times a dog has barked was a function of the time, in seconds, after its owner tied its leash to a post and left. Less than 3 minutes after he left, the owner returned, untied the leash, and walked away with the dog.
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Could each value be an input of the function? Be prepared to explain your reasoning.
15
\(84\frac12\)
300
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Could each value be an output of the function? Be prepared to explain your reasoning.
15
\(84\frac12\)
300
Student Response
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Activity Synthesis
Invite students to share their responses and reasoning. Highlight explanations that make a convincing case as to why values beyond 180 could not be inputs for this function and why fractional values could not be outputs.
Some students may argue that 300 could be an input because "300 seconds after the dog's owner walked away" is an identifiable moment, even though the dog and its owner have walked away and may no longer be near the post. Acknowledge that this is a valid point, and that it highlights the need for a function to be more specifically defined in terms of when it "begins" and "ends." If time permits, solicit some ideas on how this could be done.
Tell students that, in this lesson, they will think more about values that make sense as inputs and outputs of functions.
10.2: Card Sort: Possible or Impossible? (20 minutes)
Activity
Students continue to think about reasonable input values for functions based on the situation that they represent. They are given three functions and a set of cards containing rational values. For each function, they determine which values make sense as inputs and why. The idea of domain of a function is then introduced.
Each blackline master contains two sets of cards. Here are the numbers on the cards for your reference and planning:
- -3
- 9
- \(\frac35\)
- 15
- 0.8
- 4
- 0
- \(\frac{25}{4}\)
- 0.001
- -18
- 6.8
- 72
As students sort the cards and discuss their thinking in groups, listen for their reasons for classifying a number one way or another. Identify students whose can correctly and clearly articulate why certain numbers are or are not possible inputs.
Launch
Arrange students in groups of 2–4. Give each group a set of cards from the blackline master.
For each function defined in their activity statement, ask students to sort the cards into two groups, "possible inputs" or "impossible inputs," based on whether or not the function could take the number on the card as an input. Clarify that the cards will get sorted three times (once for each function), so students should record their sorting results for one function before moving on to the next function.
Consider asking groups to pause after sorting possible inputs for the first function and to discuss their decisions with another group. If the two groups disagree on where a number belongs, they should discuss until they reach an agreement, and then continue with the rest of the activity.
Some students may be unfamiliar with camps, and may not know that other units besides Fahrenheit and Celsius are used to measure temperature. Provide a brief orientation, if needed.
Design Principle(s): Support sense-making
Student Facing
Your teacher will give you a set of cards that each contain a number. Decide whether each number is a possible input for the functions described here. Sort the cards into two groups—possible inputs and impossible inputs. Record your sorting decisions.
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The area of a square, in square centimeters, is a function of its side length, \(s\), in centimeters. The equation \(A(s) = s^2\) defines this function.
- Possible inputs:
- Impossible inputs:
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A tennis camp charges $40 per student for a full-day camp. The camp runs only if at least 5 students sign up, and it limits the enrollment to 16 campers a day. The amount of revenue, in dollars, that the tennis camp collects is a function of the number of students that enroll.
The equation \(R(n) = 40n\) defines this function.
- Possible inputs:
- Impossible inputs:
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The relationship between temperature in Celsius and the temperature in Kelvin can be represented by a function \(k\). The equation \(k(c) = c + 273.15\) defines this function, where \(c\) is the temperature in Celsius and \(k(c)\) is the temperature in Kelvin.
- Possible inputs:
- Impossible inputs:
Student Response
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Activity Synthesis
Invite students to share their sorting results. Record and display for all to see the values students considered possible and impossible inputs for each function. Discuss any remaining disagreements students might have about particular values.
Tell students that we call the set of all possible input values of a function the domain of the function.
Ask students: "How would you describe the domain for each function?" Record and display the description that students give for each function, making sure that the descriptions are complete.
Students may not know that \(0^\circ K\) or \(\text-273.15 ^\circ C\) is absolute zero temperature, or a temperature that is agreed upon as the lowest possible temperature. Consider sharing this information with them as they describe the domain of function \(k\).
- Area: \(s\), the input of function \(A\) can be any value equal to or greater than 0 (\(s \geq 0\)). The side length can be 0 or any positive number, including irrational numbers. There may be a debate over whether 0 is a possible length of a square. Either side of the debate should be accepted as long as the connection between the input and the side length of a square is made correctly.
- Tennis camp: \(n\), the input of function \(R\) can be any whole-number value that is at least 5 and at most 16 (\(5 \leq n \leq 16\)). The number of campers cannot be fractional.
- Temperature: \(c\), the input of function \(k\) can be any value that is greater than -273.15 (\(\text- 273.15<c< \infty\)).
10.3: What about the Outputs? (10 minutes)
Activity
Earlier, students learned that the domain of a function refers to the set of all possible inputs. In this activity, students are introduced to the range of a function and examine it in terms of a situation. They begin to consider how the domain and range of a function are related to the features of its graph.
Launch
Keep students in groups of 2–4. Give students a few minutes of quiet work time, and then a moment to share their responses with their group. Leave a few minutes for whole-class discussion.
Supports accessibility for: Organization; Attention; Social-emotional skills
Student Facing
In an earlier activity, you saw a function representing the area of a square (function \(A\)) and another representing the revenue of a tennis camp (function \(R\)). Refer to the descriptions of those functions to answer these questions.
- Here is a graph that represents function \(A\), defined by \(A(s) = s^2\), where \(s\) is the side length of the square in centimeters.
- Name three possible input-output pairs of this function.
- Earlier we describe the set of all possible input values of \(A\) as “any number greater than or equal to 0.” How would you describe the set of all possible output values of \(A\)?
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Function \(R\) is defined by \(R(n) = 40n\), where \(n\) is the number of campers.
- Is 20 a possible output value in this situation? What about 100? Explain your reasoning.
- Here are two graphs that relate number of students and camp revenue in dollars. Which graph could represent function \(R\)? Explain why the other one could not represent the function.
- Describe the set of all possible output values of \(R\).
Student Response
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Student Facing
Are you ready for more?
If the camp wishes to collect at least $500 from the participants, how many students can they have? Explain how this information is shown on the graph.
Student Response
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Anticipated Misconceptions
Some students may mistakenly associate the domain and range of a function with the horizontal and vertical values that are visible in a graphing window, or with the upper and lower limits of the scale of each axis on a coordinate plane. For example, they may think that the range of the area function, \(A\), includes only values from 0 to 50 because the scale on the vertical axis goes from 0 to 50. Ask these students if it is possible to use a different scale on each axis or, if the function is graphed using technology, to adjust the graphing window. Clarify that the domain and range should be considered in terms of a situation rather than the graphing boundaries.
Activity Synthesis
Invite students to share their descriptions of the possible outputs for each function. Explain that we call the set of all possible output values of a function the range of the function. Emphasize that the range of a function depends on its domain (or all possible input values).
- For the area of the square, the range—all the possible values of \(A(s)\)—includes all numbers that are at least 0.
- For the revenue of the tennis camp, the range—all the possible values of \(R(n)\)—includes positive multiples of 40 that are at least 200 and at most 640.
Next, focus the discussion on function \(R\).
Ask students to explain which values could or could not be the outputs of \(R\) and which of the two graphs represent the function. Clarify that although the graph showing only points more accurately reflects the domain and range of the function, plotting those points could be pretty tedious. We could use a line graph to represent the function, as long as we specify or are clear that only whole numbers are in the domain and only multiples of 20 are in the range.
If time permits, draw students' attention to the temperature function they saw in an earlier activity, defined by \(k(c) = c + 273.15\). It gives the temperature in Kelvin as a function of the temperature in Celsius, \(c\). Ask students:
- "What values are in the domain of this function?" (The domain includes any value that is at least -273.15, the lowest possible temperature in Celsius, or greater)
- "What about the range?" (The range includes any value that is at least 0, the lowest possible temperature in Kelvin, or greater.)
Design Principle(s): Optimize output (for explanation); Maximize meta-awareness
10.4: What Could Be the Trouble? (10 minutes)
Optional activity
Previously, students made sense of the domain of functions in concrete contexts. This optional activity is an opportunity to reason about domain more abstractly. Students evaluate an expression that defines a function at some values of input and notice a value that produces an undefined output. They graph the function to examine its behavior, and then think about how to describe the domain of the function.
Launch
Provide access to graphing technology.
Supports accessibility for: Organization; Conceptual processing; Attention
Student Facing
Consider the function \(f(x)=\dfrac {6}{x-2}\).
To find out the sets of possible input and output values of the function, Clare created a table and evaluated \(f\) at some values of \(x\). Along the way, she ran into some trouble.
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Find \(f(x)\) for each \(x\)-value Clare listed. Describe what Clare’s trouble might be.
\(x\) -10 0 \(\frac12\) 2 8 \(f(x)\) - Use graphing technology to graph function \(f\). What do you notice about the graph?
- Use a calculator to compute the value you and Clare had trouble computing. What do you notice about the computation?
- How would you describe the domain of function \(f\)?
Student Response
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Student Facing
Are you ready for more?
Why do you think the graph of function \(f\) looks the way it does? Why are there two parts that split at \(x=2\), with one curving down as it approaches \(x=2\) from the left and the other curving up as it approaches \(x=2\) from the right?
Evaluate function \(f\) at different \(x\)-values that approach 2 but are not exactly 2, such as 1.8, 1.9, 1.95, 1.999, 2.2, 2.1, 2.05, 2.001, and so on. What do you notice about the values of \(f(x)\) as the \(x\)-values get closer and closer to 2?
Student Response
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Activity Synthesis
Display a graph of the function for all to see. Invite students to share their observations of the behavior of function \(f\) based on the completed table and their graph. Solicit their ideas on what the problem might be with this function.
If no students mentioned division by 0 as the issue, bring this up. Ask questions such as:
- "What happens when we divide a number by 0?" (The result is undefined.)
- "In the expression \(\dfrac {6}{x-2}\), what value or values of \(x\) would result in a denominator of 0?" (Only 2 gives a denominator of 0.)
- "If 2 does not produce an output, is it a possible input for \(f\)?" (No)
Highlight that the domain of \(f\) includes all numbers except 2.
Lesson Synthesis
Lesson Synthesis
Tell students that function \(q\) gives the number of minutes a person sleeps as a function of the number of hours they sleep in a 24-hour period. Display a graphic organizer such as shown.
| in the domain? | in the range? | |
|---|---|---|
| negative values | ||
| 0 | ||
| values less than 1 | ||
| 24 | ||
| 25 | ||
| 60 | ||
| fractions | ||
| values greater than 480 | ||
| 1,500 |
Ask students to decide whether each value or set of values described in the first column could be in the domain and in the range of the function. They should be prepared to explain their decisions (some of which may depend on the assumptions they made about the situation).
Once the class completes the organizer (an example is shown here), give students a moment to come up with a holistic description of the domain and range of this function.
| in the domain? | in the range? | |
|---|---|---|
| negative values | no | no |
| 0 | yes | yes |
| values less than 1 | yes | yes |
| 24 | yes | yes |
| 25 | no | yes |
| 60 | no | yes |
| fractions | yes | yes |
| values greater than 480 | no | yes |
| 1,500 | no | no |
10.5: Cool-down - Community Service (5 minutes)
Cool-Down
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Student Lesson Summary
Student Facing
The domain of a function is the set of all possible input values. Depending on the situation represented, a function may take all numbers as its input or only a limited set of numbers.
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Function \(A\) gives the area of a square, in square centimeters, as a function of its side length, \(s\), in centimeters.
- The input of \(A\) can be 0 or any positive number, such as 4, 7.5, or \(\frac{19}{3}\). It cannot include negative numbers because lengths cannot be negative.
- The domain of \(A\) includes 0 and all positive numbers (or \(s \geq 0\)).
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Function \(q\) gives the number of buses needed for a school field trip as a function of the number of people, \(n\), going on the trip.
- The input of \(q\) can be 0 or positive whole numbers because a negative or fractional number of people doesn’t make sense.
- The domain of \(q\) includes 0 and all positive whole numbers. If the number of people at a school is 120, then the domain is limited to all non-negative whole numbers up to 120 (or \(0 \leq n \leq 120\)).
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Function \(v\) gives the total number of visitors to a theme park as a function of days, \(d\), since a new attraction was open to the public.
- The input of \(v\) can be positive or negative. A positive input means days since the attraction was open, and a negative input days before the attraction was open.
- The input can also be whole numbers or fractional. The statement \(v(17.5)\) means 17.5 days after the attraction was open.
- The domain of \(v\) includes all numbers. If the theme park had been opened for exactly one year before the new attraction was open, then the domain would be all numbers greater than or equal to -365 (or \(d \geq \text-365\)).
The range of a function is the set of all possible output values. Once we know the domain of a function, we can determine the range that makes sense in the situation.
- The output of function \(A\) is the area of a square in square centimeters, which cannot be negative but can be 0 or greater, not limited to whole numbers. The range of \(A\) is 0 and all positive numbers.
- The output of \(q\) is the number of buses, which can only be 0 or positive whole numbers. If there are 120 people at the school, however, and if each bus could seat 30 people, then only up to 4 buses are needed. The range that makes sense in this situation would be any whole number that is at least 0 and at most 4.
- The output of function \(v\) is the number of visitors, which cannot be fractional or negative. The range of \(v\) therefore includes 0 and all positive whole numbers.