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.

1. The area of a square, in square centimeters, is a function of its side length, , in centimeters. The equation defines this function.

   1. Possible inputs:
   2. Impossible inputs:

2. 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 defines this function.

   1. Possible inputs:
   2. Impossible inputs:

3. The relationship between temperature in degrees Celsius and the temperature in kelvin can be represented by a function . The equation defines this function, where is the temperature in degrees Celsius and is the temperature in kelvin.

   1. Possible inputs:
   2. Impossible inputs:

In an earlier activity, you saw a function representing the area of a square (function ) and another representing the revenue of a tennis camp (function ). Refer to the descriptions of those functions to answer these questions.

1. Here is a graph that represents function , defined by , where is the side length of the square in centimeters.

   

   A graph with origin O. The horizontal axis, side length, centimeters, scale from 0 to 8 by 0.5’s. The vertical axis, area, square centimeters, scale from 0 to 50 by 25s. A curve, trending up and to the right, passes through the points 0 comma 0, 4 comma 16, and 6 comma 36.

   1. Name three possible input-output pairs of this function.
   2. Earlier we described the set of all possible input values of as “any number greater than or equal to 0.” How would you describe the set of all possible output values of ?

2. Function is defined by , where is the number of campers.

   1. Is 20 a possible output value in this situation? What about 100? Explain your reasoning.
   2. Here are two graphs that relate number of students and camp revenue in dollars. Which graph could represent function ? Explain why the other one could not represent the function.

      

      A graph with origin O. The horizontal axis, number of students, scale from 0 to 20 by 1s. The vertical axis, amount in dollars, scale from 0 to 700 by 50s. A line, trending up and to the right, passes through the points 0 comma 0, 5 comma 200, and 15 comma 600.

      

      A graph with origin O. The horizontal axis, number of students, scale from 0 to 20 by 1s. The vertical axis, amount in dollars, scale from 0 to 700 by 50s. Points are marked at 5 comma 200, 6 comma 240, 7 comma 280, 8 comma 320, 9 comma 360, 10 comma 400, 11 comma 440, 12 comma 480, 13 comma 520, 14 comma 560, 15 comma 600, 16 comma 640.

   3. Describe the set of all possible output values of .

Consider the function . 

To find out the sets of possible input and output values of the function, Clare created a table and evaluated at some values of . Along the way, she ran into some trouble.

1. Find  for each -value Clare listed. Describe what Clare’s trouble might be.

   	-10	0		2	8

2. Use graphing technology to graph function . What do you notice about the graph?
3. Use a calculator to compute the value you and Clare had trouble computing. What do you notice about the computation?
4. How would you describe the domain of function ?