The goal for this activity is for students to define the domain of a function by examining what inputs are possible for various functions. The activity also gives students different types of numbers that they may consider when examining domains of functions in the future.

Students sort different values as inputs during this activity. A sorting task gives students opportunities to analyze representations, statements, and structures closely and make connections (MP2, MP7).

Monitor for different ways groups choose to categorize the values, but especially for categories that distinguish between different number types such as rationals, positives, negatives, and integers. 

Each blackline master contains two sets of cards. Here are the numbers on the cards for your reference and planning:

• -3
• 9
• 15
• 0.8
• 4
• 0
• 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 who can correctly and clearly articulate why certain numbers are or are not possible inputs.

Select groups 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 or , is absolute zero, or the temperature that is agreed upon as the lowest possible temperature. Consider sharing this information with them as they describe the domain of function .

• Area: , the input of function can be any value equal to or greater than 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: , the input of function can be any whole-number value that is at least 5 and at most 16 ( ). The number of campers cannot be fractional.
• Temperature: , the input of function can be any value that is greater than -273.15 (\(\text- 273.15).

Some students may mistakenly associate the possible inputs and outputs 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. While that is good practice in situations in which it is difficult to determine what would happen outside of what is graphed, if we understand the situation more clearly, we can sometimes say more about what is possible. For example, students may think that the set of possible outputs of the area function, , includes values only 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 possible inputs and outputs  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 —includes all numbers that are at least 0.
• For the revenue of the tennis camp, the range—all the possible values of —includes positive multiples of 40 that are at least 200 and at most 640.

Next, focus the discussion on function .

Ask students to explain which values could or could not be the outputs of 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 . It gives the temperature in kelvin as a function of the temperature in degrees Celsius, . 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 degrees 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.)

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. Rational functions are explored in more depth in a later course.

Display a graph of the function for all to see. Invite students to share their observations of the behavior of function 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 , what value or values of 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 ?" (No.)

Highlight that the domain of includes all numbers except 2.