Finding Domain and Range Using a Graph

4.13.3 Finding Domain and Range Using a Graph

4.13.3 • Finding Domain and Range Using a Graph

Activity

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.

A graph shows the height (feet) of a bouncing ball over time (seconds), with labeled points marking the height and time at each bounce: (0,6), (0.28,4.3), (0.54, 0), (1.03,3.5), (0.74,2.2), (1.865,0), (1.48,0), (2.25,0), and (3.0,0).

1. Find $H (0)$ . What point is this on the graph? What does it represent?

Compare your answer:

$H (0) = 6$ . It tells us the ball was dropped from a height of 6 feet at $t = 0$ .

2. Solve $H (x) = 0$ .

Compare your answer:

$x = 0.54$ , $x = 1.48$ , and $x = 2.25$ are three possible solutions, but there are others. Each one represents the time when the ball hit the ground.

3. Describe the domain of the function.

Compare your answer:

The domain includes all values between 0 and the time when the ball stopped moving, which, based on the graph, is about 3.8.

4. Describe the range of the function.

Compare your answer:

The range includes all values between 0 and 6 feet, the height from which the ball was dropped.

Are you ready for more?

Extending Your Thinking

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.

Compare your answer:

The maximum height, in feet, that the ball reaches after $n$ bounces as a function of the number of bounces.

2. Sketch a graph of this function.

A blank Cartesian graph with horizontal (x-axis) and vertical (y-axis) black arrows, both starting from the origin marked as O in the bottom left corner. No data or labels are shown.

Compare your answer:

A scatter plot showing the maximum height in feet versus the number of bounces. As the number of bounces increases from 1 to 4, the maximum height decreases from about 6 to 0.5 feet.

3. Describe a function whose range includes only integers. Be sure to specify the units.

Compare your answer:

The number of times the ball has hit the ground as a function of time, in seconds, since the ball was dropped.

4. Sketch a graph of this function.

Compare your answer:

A step graph shows the number of times a ball hits the ground over time. The count increases at regular intervals, starting at 0 and reaching 4 by 4 seconds. Both axes are labeled.

Video: Interpreting Function Notation

Watch the following video to learn more about function notation.

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Self Check

The graph below shows the number of customers, $C$ , in a restaurant $x$ hours after 8 a.m.

What does $C(14)$ equal, and what does it mean?

A GRAPHED FUNCTION, WHERE NUMBER OF CUSTOMERS IS A FUNCTION OF THE NUMBER OF HOURS AFTER 8 AM. THE GRAPH INCREASES, DECREASES, INCREASES, AND THEN DECREASES AGAIN.

$C(14)=0$ means there have not been any customers over 14 hours.
$C(14)=0$ and means there are 14 customers at 8 a.m.
$C(14)=0$ and is when the restaurant closes and has no more customers around 10 p.m.
$C(14)=0.5$ and means there are 14 customers around 8:30.

Additional Resources

Domain and Range from Graphs

The graph below shows the distance Carl drives from home, $D (t)$ , when $t$ is time in seconds.

Line graph showing distance from home (meters) over time (seconds). The distance increases, peaks at 285 meters around 180 seconds, then decreases back to zero by 400 seconds. Grid lines in background.

1. What is the domain of the graph?

The domain is $x$ -values that are at least 0 and a maximum of 400.

2. What is the range of the graph?

The range is the $y$ -values that are at least 0 and a maximum of 285.

3. What are the values of $x$ where $D (x) = 0$ , and what do they mean?

$D (0) = 0$ and $D (400) = 0$ , and this is when Carl left home and then arrived home, so the distance was 0.

Try it

Domain and Range from Graphs

What are the values for the domain and range of the graph below?

Line graph showing voltage (in volts) versus time (in seconds). The voltage rises, peaks at 3V at 2s, falls to 1V at 5s, stays flat, then rises to 2V at 9s and back to 1V at 10s..

Here is how to find the domain and range of the graph:

The domain includes the $x$ -values or inputs of the graph. The domain is from at least 0 to at most 10.

The range includes the $y$ -values or outputs of the graph. The range is from at least 1 to at most 3.