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<h4>Activity (10 minutes)</h4>
<p>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.</p>
<h4>Launch</h4>
<p>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.</p>
<br>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for Students with Disabilities</p>
<p class="os-raise-extrasupport-name">Engagement: Provide Access by Recruiting Interest</p>
</div>
<div class="os-raise-extrasupport-body">
<p>
Leverage choice around perceived challenge. Invite students to analyze either the area function or the revenue
function. </p>
<p class="os-raise-text-italicize">Supports accessibility for: Organization; Attention; Social-emotional skills</p>
</div>
</div>
<br>
<h4>Student Activity </h4>
<p>In an earlier activity, you saw a function representing the area of a square (function \(A\)) and another
representing the revenue of a tennis camp (function \(R\)). Refer to the descriptions of those functions to answer
these questions.</p>
<ol class="os-raise-noindent">
<li> Here is a graph that represents function \(A\), defined by \(A(s)=s^2\), where \(s\) is the side length of the
square in centimeters. <br>
<br>
<img alt="A graph. " src="https://k12.openstax.org/contents/raise/resources/76576c471fa89bdb2f5e278ace7605136df739d5">
</li>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" type="a">
<li> Name three possible input-output pairs of this function. Enter your answers as sets of ordered pairs. </li>
<p> <strong>Answer: </strong>\(A(2)=4\), \(A(3)=9\), and \(A(7)=49\), or \((2,4)\), \((3,9)\), \((7,49)\)</p>
<li> Earlier, we described the set of all possible input values of \(A\) as “any number greater than or equal
to 0.” How would you describe the set of all possible output values of \(A\)? </li>
<p> <strong>Answer: </strong> The outputs of \(A\) are also all numbers greater than or equal to 0. </p>
</ol>
</ol>
<ol class="os-raise-noindent" start="2">
<li> Function \(R\) is defined by \(R(n)=40n\), where \(n\) is the number of campers.
<ol class="os-raise-noindent" type="a">
<li> Is 20 a possible output value in this situation? What about 100? Be prepared to show your reasoning. </li>
<p> <strong>Answer: </strong> No, both 20 and 100 cannot be outputs because the input (the number of campers)
cannot be fractions. </p>
<li> Here are two graphs that relate number of students and camp revenue in dollars. Which graph could represent
function \(R\)? Explain why the other one could not represent the function.<br>
<br>
<img alt="A graph. " src="https://k12.openstax.org/contents/raise/resources/4ef73b4c27d99682a72671f9517d57581e0d3589"> <br>
<br>
<img alt="A graph. " src="https://k12.openstax.org/contents/raise/resources/8e212f8010016b9ef2ed559709268813ad1477b2">
</li>
</ol>
</li>
<p> <strong>Answer: </strong> The second graph could represent function \(R\). The first graph could not represent the
function \(R\) because it includes all points for \(n\)-values between 0 and 5, \(n\)-values greater than 16, and
fractional \(n\)-values. All of these \(n\)-values don’t apply in this situation. </p>
</ol>
<ol class="os-raise-noindent">
<ol class="os-raise-noindent" start="3" type="a">
<li> Describe the set of all possible output values of \(R\). </li>
</ol>
<p> <strong>Answer: </strong> The outputs of \(R\) are all multiples of 40 between 200 and 640 (200, 240, 280, and so
on). </p>
</ol>
<h4>Video: Identifying Outputs of a Function</h4>
<p>Watch the following video to learn more about identifying outputs of a function.</p>
<div class="os-raise-d-flex-nowrap os-raise-justify-content-center">
<div class="os-raise-video-container"><video controls="true" crossorigin="anonymous">
<source src="https://k12.openstax.org/contents/raise/resources/21adf0146df228d483924077672ad2140a3bb95d">
<track default="true" kind="captions" label="On" src="https://k12.openstax.org/contents/raise/resources/e0decf3ffc5d4cbc1b14b2110a48743a7ffa6739" srclang="en_us">
https://k12.openstax.org/contents/raise/resources/21adf0146df228d483924077672ad2140a3bb95d
</video></div>
</div>
<br>
<br>
<h4>Student Facing Extension</h4>
<h5>Are you ready for more?</h5>
<p>If the camp wishes to collect at least $500 from the participants, how many students can they have? Explain how this
information is shown on the graph.</p>
<h4>Extension Student Response</h4>
<p>They will need at least 13 students to collect $500 or more, so the possibilities are 13, 14, 15, and 16. On the
graph, there are only 4 dots that lie on or above the horizontal line representing $500.</p>
<h4>Anticipated Misconceptions</h4>
<p>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.</p>
<h4>Activity Synthesis</h4>
<p>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).</p>
<ul>
<li> For the area of the square, the range—all the possible values of \(A(s)\)—includes all numbers that
are at least 0. </li>
<li> 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. </li>
</ul>
<p>Next, focus the discussion on function \(R\).</p>
<p>Ask students to explain which values could or could not be the outputs of \(R\) and which of the two graphs
represents 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.</p>
<p>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:</p>
<ul>
<li>“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.) </li>
<li>“What about the range?” (The range includes any value that is at least 0, the lowest possible
temperature in Kelvin, or greater.) </li>
</ul>
<div class="os-raise-extrasupport">
<div class="os-raise-extrasupport-header">
<p class="os-raise-extrasupport-title">Support for English Language Learners</p>
<p class="os-raise-extrasupport-name">MLR 3 Clarify, Critique, Correct: Reading, Writing, Speaking</p>
<p>
</div>
<div class="os-raise-extrasupport-body">
Before students share their descriptions of the possible output values of \(A\), present an incorrect response and
explanation. For example, “The outputs of \(A\) are numbers from 0 to 50 because I looked on the vertical axis
and saw that the graph reaches up to 50.” Ask students to identify the error, critique the reasoning, and
write a correct explanation. As students discuss with a partner, monitor for students who clarify that the output
values are not restricted by the graphing boundaries shown. This helps students evaluate, and improve upon, the
written mathematical arguments of others as they discuss the range of a function.</p>
<p class="os-raise-text-italicize">Design Principle(s): Optimize output (for explanation); Maximize meta-awareness</p>
</div>
</div>
<br>
<h3>4.12.3: Self Check</h3>
<p class="os-raise-text-bold"><em>After the activity, students will answer the following question to check their understanding of the
concepts explored in the activity.</em></p>
<p class="os-raise-text-bold">QUESTION:</p>
<p>Looking at the graph below, which statement about the values graphed is true?</p>
<p><img alt="SCATTER PLOT WITH POINTS AT (2, 7), (3, 9), (4, 11), (5, 13), (6, 15), (7, 17), AND (8, 19)." src="https://k12.openstax.org/contents/raise/resources/94a38b96d0df6fd05f92cff8642a934fc7ffc7b2" width="300"></p>
<table class="os-raise-textheavytable">
<thead>
<tr>
<th scope="col">Answers</th>
<th scope="col">Feedback</th>
</tr>
</thead>
<tbody>
<tr>
<td>
The largest output value is 8.
</td>
<td>
Incorrect. Let’s try again a different way: When graphed, the largest input value is 8 since it is an
\(x\)-value in the point \((8,19)\). The answer is the largest output value is 19.
</td>
</tr>
<tr>
<td>
The largest output value is 19.
</td>
<td>
That’s correct! Check yourself: The largest \(y\)-value graphed is 19, so that is the largest output.
</td>
</tr>
<tr>
<td>
The largest input value is 19.
</td>
<td>
Incorrect. Let’s try again a different way: Since 19 in the point \((8,19)\) is a \(y\)-coordinate, it
is an output value. The answer is the largest output value is 19.
</td>
</tr>
<tr>
<td>
All odd numbers are outputs.
</td>
<td>
Incorrect. Let’s try again a different way: While the output values are odd numbers, the numbers in the
outputs do not begin until 7. The answer is the largest output value is 19.
</td>
</tr>
</tbody>
</table>
<br>
<h3>4.12.3: Additional Resources</h3>
<p class="os-raise-text-bold"><em>The following content is available to students who would like more support based on their experience with
the self check. Students will not automatically have access to this content, so you may wish to share it with
those who could benefit from it. </em></p>
<h4>Naming Input-Output Pairs</h4>
<p>A graph is yet another way that a relation can be represented. The set of ordered pairs of all the points plotted is
the relation. The set of all \(x\)-coordinates, or the input values, is the domain of the relation, and the set of all
\(y\)-coordinates, or outputs, is the range.</p>
<p>Recall a coordinate is written \((x, y)\).</p>
<p>Name the input-output pairs for the graph below:</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/57ace179bf0134cf2d8555e9ce779dae0fd467c8" width="300">
</p>
<p>The input values are the \(x\)-values, and the output values are y-values. \((1,3) (2,6) (3,12) (4,24)\)</p>
<p>This means the domain for the graphed relation is \({1, 2, 3, 4}\) and the range is \({3, 6, 12, 24}\)</p>
<h4>Try It: Naming Input-Output Pairs</h4>
<p>Name the input-output pairs of the graph below:</p>
<p><img src="https://k12.openstax.org/contents/raise/resources/24ef66d8ee5e588cdb682ef865db1e5ccb4e127c" width="300">
</p>
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work.</p>
<h5>Solution</h5>
<p>Here is how to write the input-output pairs:</p>
<p>\((1, 4.5)\) \((2, 9)\) \((3, 18)\) \((4,36)\)</p>
<h4>Using Inequalities to Determine Range </h4>
<p>The domain and range of a function describe the set of possible input values (domain) and output values (range) for that function. The range represents the values that the function can output. Inequalities can be used to represent the range of a function. Remember to consider the nature of the function, its properties, and any specific restrictions when determining the range using inequalities. Graphing the function can often provide visual clues to help confirm your analysis.</p>
<br>
<p>Use the following information to determine the range of a function using inequalities:</p>
<ul>
<li>Analyze the behavior and possible output values of the function, usually denoted as \(f(x)\).</li>
<li>Consider the range of the function based on its graph or algebraic properties.</li>
<li>Identify the minimum and maximum possible values that the function can reach.</li>
</ul>
<br>
<p>Use the following information to express the range of a function using inequalities:
<ul>
<li>Use interval notation or set notation to represent the valid output values for the function.</li>
<li> Use inequalities to express the minimum and maximum possible values.For example, if the function's graph never goes below −3 and can reach any positive value, you would write \((−3 < < +∞)\) as an inequality. Try It Solutions </li>
</ul>
<!-- BEGIN GRAY BOX STYLING-->
<br>
<div class="os-raise-graybox">
<p> <strong> Special cases </strong></p>
<hr>
<p>Consider any specific exclusions or additional conditions mentioned in the function's definition.</p>
<p>For instance, if a function has a denominator and \(x\) cannot equal certain values, denote those exclusions in the domain using the notation "\(x\) ≠ value." </p>
</div>
<br>
<!-- END GRAY BOX STYLING-->
<p><strong>Example:</strong></p>
<p>Determine the range of this function: \(p(x) = 3, x ≥ −2.\)</p>
<p>Since the function \(p(x)\) is constant and always equal to 3, the range of the function is a single value, which is 3.</p>
<p>Therefore, the range of \(p(x) = 3\), \(x ≥ −2\) is the set {3}.</p>
<br>
<h4>Try It: Determine the Range of a Function Using Inequalities</h4>
<br>
<ol class="os-raise-noindent">
<li> What is the range of the function \(f(x)=2x-3\) using inequalities.</li>
</ol>
<p><strong>Answer: </strong><br>
\(R: (−∞ < f < +∞) \)</p>
<ol class="os-raise-noindent" start="2">
<li> What is the range of the function \(g(x)=\frac{1}{(x-2)}\), \(x ≠ 2\) using inequalities.</li>
</ol>
<p><strong>Answer: </strong><br>
\(R: (−∞ > g > 0) \cup (0 < g< +∞) \)indicating that the function can take any value except 0</p>
</p>