ee30b4dc-2278-40b8-bfc7-830e7c775c0d.html

<h4>Activity (20 minutes)</h4>
<p>Students continue to think about reasonable input values for functions based on the situation that they represent.
  They are given three functions and a set of cards containing rational values. For each function, they determine which
  values make sense as inputs and why. The idea of the domain of a function is then introduced.</p>
<p>Each blackline master contains two sets of cards. Here are the numbers on the cards for your reference and planning:
</p>
<ul>
  <li>&ndash;3 </li>
  <li> 9 </li>
  <li>\(\frac35\) </li>
  <li> 15 </li>
  <li> 0.8 </li>
  <li> 4 </li>
  <li> 0 </li>
  <li> \(\frac{25}4\) </li>
  <li> 0.001 </li>
  <li>&ndash;18 </li>
  <li> 6.8 </li>
  <li> 72 </li>
</ul>
<p>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.</p>
<h4>Launch</h4>
<p>Arrange students in groups of 2&ndash;4. Give each group a set of cards from the <a
    href="https://k12.openstax.org/contents/raise/resources/0e6a998a9a2112809cdc487e3fcfed51de8847f7" target="_blank">blackline
    master</a>.</p>
<p>For each function defined in their activity statement, ask students to sort the cards into two groups,
  &ldquo;possible inputs&rdquo; and &ldquo;impossible inputs,&rdquo; based on whether the function could take the number
  on the card as an input. Clarify that the cards will get sorted three times (once for each function), so students
  should record their sorting results for one function before moving on to the next function.</p>
<p>Consider asking groups to pause after sorting possible inputs for the first function and to discuss their decisions
  with another group. If the two groups disagree on where a number belongs, they should discuss until they reach an
  agreement, and then they should continue with the rest of the activity.</p>
<p>Some students may be unfamiliar with camps, and they may not know that other units besides Fahrenheit and Celsius are
  used to measure temperature. Provide a brief orientation, if needed.</p>

  
<h4>Support for English Language Learners</h4>
<div class="os-raise-graybox">

Action and Expression: Provide Access for Physical Action
Supports accessibility for: Organization; Conceptual processing; Attention</div>
<br>


  <p>After sorting possible inputs for the first function, provide the class with the following sentence frames to help
    groups respond to each other: &ldquo;_____ is a possible/impossible input because . . .&rdquo; and &ldquo;I
    agree/disagree because . . . .&rdquo; When monitoring discussions, re-voice student ideas to demonstrate
    mathematical language. This will help students listen and respond to each other as they explain how they sorted the
    cards.</p>
</div>
<br>
<h4>Student Activity</h4>
<p>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.</p>
<ol class="os-raise-noindent">
  <li> The area of a square, in square centimeters, is a function of its side length, \(s\), in centimeters. The
    equation \(A(s) =s^2\) defines this function. </li>
  <ol class="os-raise-noindent" type="a">
    <li> Possible inputs: </li>
    <li> Impossible inputs: </li>
  </ol>
</ol>
<p> <strong>Answer: </strong> The area of a square, in square centimeters, is a function of its side length, s, in
  centimeters. The equation \(A(s) =s^2\) defines this function. </p>
<ol class="os-raise-noindent" type="a">
  <li> Possible inputs: \(9\), \(\frac35\), \(15\), \(0\), \(0.8\), \(4\),
    \(\frac{25}4\)
    , \(0.001\), \(6.8\), \(72\) </li>
  <li> Impossible inputs: &ndash;3, &ndash;18 </li>
</ol>
<ol class="os-raise-noindent" start="2">
  <li> A tennis camp charges $40 per student for a full-day camp. The camp only runs 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 who enroll.
    The equation \(R(n)=40n\) defines this function.
    <ol class="os-raise-noindent" type="a">
      <li> Possible inputs: </li>
      <li> Impossible inputs: </li>
    </ol>
  </li>
</ol>
<p class="os-raise-text-bold">Answer:</p>
<ol class="os-raise-noindent" type="a">
  <li> Possible inputs: 9, 15 </li>
  <li> Impossible inputs: \(-3\), \(-18\), \(\frac35\), \(0.8\), \(0\), \(\frac{25}4\), \(0.001\), \(4\), \(6.8\),
    \(72\) </li>
</ol>
<ol class="os-raise-noindent" start="3">
  <li> The relationship between temperature in Celsius and the temperature in Kelvin can be represented by a function
    \(k\). The equation \(k(c)=c+273.15\) defines this function, where \(c\) is the temperature in Celsius and \(k(c)\)
    is the temperature in Kelvin.
    <ol class="os-raise-noindent" type="a">
      <li> Possible inputs: </li>
      <li> Impossible inputs: </li>
    </ol>
  </li>

  <p class="os-raise-text-bold">Answer:</p>
  <ol class="os-raise-noindent" type="a">
    <li> Possible inputs: All values </li>
    <li> Impossible inputs: No values </li>
  </ol>

  <p>Write this definition: The domain of a function is the set of all of its possible input values.</p>
</ol>
<!-- <p>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.</p> -->

<h4>Activity Synthesis</h4>
<p>Students may not know that 0&deg;K or &ndash;273.15&deg;C is absolute zero temperature, or a temperature that is
  agreed upon as the lowest possible temperature. Consider sharing this information with them as they describe the
  domain of function \(k\).</p>
<ul>
  <li> Area: \(s\), the input of function \(A\), can be any value equal to or greater than 0: \(s\geq0\). 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 the side 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. </li>
  <li> Tennis camp: \(n\), the input of function \(R\), can be any whole-number value that is at least 5 and at most 16:
    \(5\leq n\leq16\). The number of campers cannot be fractional. </li>
  <li> Temperature: \(c\), the input of function \(k\), can be any value that is greater than &ndash;273.15. </li>
</ul>
<h3>4.12.2: 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>Jose created a function \(W(h)\) for the weight, \(W\), of his family members given their height, \(h\), in inches.</p>
        <p>Which of the following provides a group of possible inputs for the function?</p>
        <table class="os-raise-textheavytable">
          <thead>
            <tr>
              <th scope="col">Answers</th>
              <th scope="col">Feedback</th>
            </tr>
          </thead>
          <tbody>
            <tr>
      <td>
        \(36\), \(52\), \(-70\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: Height cannot be negative. The answer is \(47\), \(50.2\),
          \(60\frac12\).
      </td>
    </tr>
    <tr>
      <td>
       \(5\), \(40\), \(62\)
      </td>
      <td>
        Incorrect. Let&rsquo;s try again a different way: A height of 5 inches is not realistic. The answer is
          \(47\), \(50.2\), \(60\frac12\).
      </td>
    </tr>
    <tr>
      <td>
        \(47\), \(50.2\), \(60\frac12\)
      </td>
      <td>
        That&rsquo;s correct! Check yourself: Height can be expressed in fractions and decimals, and these are
          reasonable values for the input in inches.
      </td>
    </tr>
    <tr>
      <td>
       \(62\), \(-68\), \(70\frac34\)
      </td>
      <td>
       Incorrect. Let&rsquo;s try again a different way: Height cannot be negative. The answer is \(47\), \(50.2\),
          \(60\frac12\).
      </td>
    </tr>
  </tbody>
</table>
<br>
<h3>4.12.2: 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>Finding Possible Input Values</h4>

<ol class="os-raise-noindent">
  <li>Micah mows lawns every week where the function \(M\) represents the number of lawns mowed as a function of \(d\), days, modeled by the equation \(M(d)=4d\). Select <strong>three</strong> of the following that are inputs for this function.</li>
</ol>    
<ul>
  <li>&minus;2</li>
  <li>0</li>
  <li>2.5</li>
  <li>4</li>
</ul>
<p><strong>Answer:</strong> 0, 2.5, and 4. &minus;2 is incorrect because it is not possible for him to work a negative number of days.</p>
      
<ol class="os-raise-noindent" start="2">
  <li>Elena runs at a constant speed. Her distance, \(D\), in miles is a function of the time, \(t\), in minutes. Her running distance is represented by the function \(D(t)=9t\). Select <strong>three</strong> of the following that are inputs for this function.</li>
</ol>    
<ul>
  <li>&minus;7</li>
  <li>40</li>
  <li>\(\frac{2}{3}\)</li>
  <li>60.5</li>
</ul>
<p><strong>Answer:</strong> 40, 60.5, and \(\frac{2}{3}\).</p>
<p>&minus;7 is not possible because she cannot run for a negative amount of time.</p>
<p>\(\frac{2}{3}\) is possible. However, it is not always reasonable because it stands for \(\frac{2}{3}\) of one minute. It is important to always look at the entire problem to see if there are any other restrictions, like a minimum amount of time running.</p>
<p>45 - 60.5 are both possible input values because she could run for 45 minutes or 60.5 minutes.</p>

<h4>Using Inequalities to Determine Domain</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 domain represents the values that can be plugged into the function. Inequalities can be used to represent the domain. Remember to consider the nature of the function, its properties, and any specific restrictions when determining the domain using inequalities. Graphing the function can often provide visual clues and help confirm your analysis.</p>
<br>
<p>Use the following information to determine the domain of a function using inequalities:</p>
<ul>
  <li>Identify any restrictions or limitations on the input variable, usually denoted as \(x\).</li>
  <li>Look for values of \(x\) that could result in undefined or non-existent outputs.</li>
  <li>Common restrictions include square roots of negative numbers, division by zero, or values excluded by the function's definition.</li>
</ul>
<br>
<p>Use the following information to express the domain of a function using inequalities:</p>
<ul>
  <li>Use interval notation or set notation to represent the valid input values for the function.</li>
  <li>Use inequalities to express any restrictions.</li>
  <li>For example, if \(x\) cannot be negative, you would write \(x \geq 0\) or \((- \infty, 0] \cup (0, + \infty)\) in interval notation.</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 "\(x \neq value\)."</p>  
</div>
<br>
<!-- END GRAY BOX STYLING-->

<p><strong>Example</strong></p>
<p>Determine the domain of this function: \(p(x) = 3, x \geq -2\).</p>
<p>The inequality \(x \geq -2\) indicates that the function is defined for all values of \(x\) that are greater than or equal to \(-2\).</p>
<p>Therefore, the domain of \(p(x) = 3, x \geq -2\) is \(x \geq -2\), or in interval notation, \([-2, +\infty\).</p>
<br>
<h4>Try It: Determine the Domain of a Function Using Inequalities</h4>
<br>
<ol class="os-raise-noindent">
  <li>What is the domain of the function \(f(x) = 0.5x\) using inequalities, where \(x\) represents the time in seconds a turtle has walked after his race with the hare begins?</li>
</ol>     
<p><strong>Answer:</strong> \(x \geq 0\) is the domain written as an inequality. While the linear function \(f(x) = 0.5x\) has a domain of all real numbers, because the scenario describes the input as the time since the race began, it creates a set of limitations or restrictions on the domain.</p>
<ul>
  <li>The time the race starts begins at 0 seconds. In other words, it does not make sense for time to be negative. Thus, \(x\) must be greater than or equal to zero.</li>
  <li>And, because time can be measured with decimal or fractional values in addition to whole numbers, we know that \(x\) can be any real number greater than or equal to zero.</li>
</ul>
<p>The interval notation for this domain is \([0, +\infty)\). Because the race starts at 0 seconds, it is included as part of the domain.</p>

<ol class="os-raise-noindent" start="2">
  <li>What is the domain of the function \(f(x) = 3x\) using inequalities, where \(x\) represents the number of tickets sold by a dance team for a fundraising raffle?</li>
</ol>     
<p><strong>Answer:</strong> The domain written as an inequality is \(x \geq 0\), where \(x\) is a whole number. While the linear function \(f(x) = 3x\) has a domain of all real numbers, because the scenario describes the input as the number of tickets sold, it creates a set of limitations or restrictions on the domain.</p>
<ul>
  <li>The dance team can sell 0, 1, 2, or more tickets, but they can't sell a negative number of tickets. Thus, \(x\) must be greater than or equal to zero.</li>
  <li>And, because it does not make sense to sell half a raffle ticket (or any other fractional amount), we need to specify that the values of \(x\) are whole numbers.</li>
</ul>
<p>The interval notation for this domain is "\(x\) is a whole number such that \([0, +\infty)\)." Notice that the dance team could sell 0 tickets so that value is included in the domain.</p>

<ol class="os-raise-noindent" start="3">
  <li>What is the domain of the function \(g(x)=\frac{1}{(x-2)}\), \(x \neq 2\) using inequalities?</li>
</ol>     
<p><strong>Answer:</strong> \(x < 2\) or \(x > 2\) is the domain written with inequalities. To write the domain in interval notation, we use \((- \infty, 2) \cup (2, + \infty)\). Notice that the domain cannot equal 2 so it is not included in the notation.</p>