Lesson 11

Domain and Range (Part 2)

Let’s analyze graphs of functions to learn about their domain and range.

11.1: Which One Doesn't Belong: Unlabeled Graphs

Which one doesn't belong?

A

A graph with origin O. The graph is a line crossing the vertical axis and slants downward to cross the horizontal axis.

B

A graph. 

C

A graph with origin O. The graph is a horizontal line passing through the middle of the vertical axis.

D

A graph.

11.2: Time on the Swing

A child gets on a swing in a playground, swings for 30 seconds, and then gets off the swing.

  1. Here are descriptions of four functions in the situation and four graphs representing them.

    The independent variable in each function is time, measured in seconds.

    A girl on a swing. 

    Match each function with a graph that could represent it. Then, label the axes with the appropriate variables. Be prepared to explain how you make your matches.

    • Function \(h\): The height of the swing, in feet, as a function of time since the child gets on the swing
    • Function \(r\): The amount of time left on the swing as a function of time since the child gets on the swing
    • Function \(d\): The distance, in feet, of the swing from the top beam (from which the swing is suspended) as a function of time since the child gets on the swing
    • Function \(s\): The total number of times an adult pushes the swing as a function of time since the child gets on the swing

    A

    A graph with origin O. The graph is a line crossing the vertical axis and slants downward to cross the horizontal axis.

    B

    A graph. 

    ​​​​​​

    C

    A graph with origin O. The graph is a horizontal line passing through the middle of the vertical axis.

    D

    A graph.
  2. On each graph, mark one or two points that—if you have the coordinates—could help you determine the domain and range of the function. Be prepared to explain why you chose those points.
  3. Once you receive the information you need from your teacher, describe the domain and range that would be reasonable for each function in this situation.

11.3: Back to the Bouncing Ball

A tennis ball was dropped from a certain height. It bounced several times, rolled along for a short period, and then stopped. Function \(H\) gives its height over time.

Here is a partial graph of \(H\). Height is measured in feet. Time is measured in seconds.

Use the graph to help you answer the questions.

Be prepared to explain what each value or set of values means in this situation.

Nonlinear function. Time (seconds) and height (feet).
  1. Find \(H(0)\).
  2. Solve \(H(x) = 0\).
  3. Describe the domain of the function.
  4. Describe the range of the function.


In function \(H\), the input was time in seconds and the output was height in feet. 

Think about some other quantities that could be inputs or outputs in this situation.

  1. Describe a function whose domain includes only integers. Be sure to specify the units.
  2. Describe a function whose range includes only integers. Be sure to specify the units.
  3. Sketch a graph of each function.
    A blank graph. Origin O. 
    A blank graph. Origin O. 

Summary

The graph of a function can sometimes give us information about its domain and range.

Here are graphs of two functions we saw earlier in the unit. The first graph represents the best price of bagels as a function of the number of bagels bought. The second graph represents the height of a bungee jumper as a function of seconds since the jump began.

What are the domain and range of each function?

The number of bagels cannot be negative but could include 0 (no bagels bought). The domain of the function therefore includes 0 and positive whole numbers, or \(n \geq 0\).

The best price can be \$0 (for buying 0 bagels), certain multiples of 1.25, certain multiples of 6, and so on. The range includes 0 and certain positive values.

A graph. 

The domain of the height function would include any amount of time since the jump began, up until the jump is complete. From the graph, we can tell that this happened more than 70 seconds after ​​​​​​the jump began, but we don't know the exact value of \(t\).

The graph shows a maximum height of 80 meters and a minimum height of 10 meters. We can conclude that the range of this function includes all values that are at least 10 and at most 80.

A graph. 

Glossary Entries

  • domain

    The domain of a function is the set of all of its possible input values.

  • range

    The range of a function is the set of all of its possible output values.