Lesson 10

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.

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.

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.

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.

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 (\(\text- 273.15<c< \infty\)).

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.

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.

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.

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.)

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.

Provide access to graphing technology.

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.