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<h4>Activity </h4>
<span id="docs-internal-guid-180eef2b-7fff-2dde-333d-ca8fa2851b02"> Consider the functions \(f(x)=2x\) and
\(g(x)=(1.01)^x\).<br></span><br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="fde11bc0-abd9-4022-a33a-f7c1cf81e1bd"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent">
<li> Complete the table of values for the functions \(f\) and \(g\). </li>
</ol>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
\(x\)
</th>
<th scope="col">
\(f(x)\)
</th>
<th scope="col">
\(g(x)\)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
1
</td>
<td></td>
<td></td>
</tr>
<tr>
<td>
10
</td>
<td></td>
<td></td>
</tr>
<tr>
<td>
50
</td>
<td></td>
<td></td>
</tr>
<tr>
<td>
100
</td>
<td></td>
<td></td>
</tr>
<tr>
<td>
500
</td>
<td></td>
<td></td>
</tr>
</tbody>
</table>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<table class="os-raise-midsizetable">
<thead>
<tr>
<th scope="col">
\(x\)
</th>
<th scope="col">
\(f(x)\)
</th>
<th scope="col">
\(g(x)\)
</th>
</tr>
</thead>
<tbody>
<tr>
<td>
1
</td>
<td>
2
</td>
<td>
1.01
</td>
</tr>
<tr>
<td>
10
</td>
<td>
20
</td>
<td>
~1.10
</td>
</tr>
<tr>
<td>
50
</td>
<td>
100
</td>
<td>
~1.645
</td>
</tr>
<tr>
<td>
100
</td>
<td>
200
</td>
<td>
~2.7
</td>
</tr>
<tr>
<td>
500
</td>
<td>
1,000
</td>
<td>
~144.8
</td>
</tr>
</tbody>
</table>
<br>
</div>
</div>
<!--Interaction End -->
<br>
<div class="os-raise-ib-cta" data-button-text="Solution" data-fire-event="Reveal1" data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<ol class="os-raise-noindent" start="2">
<li> Based on the table of values, which function do you think grows faster? Be prepared to show your reasoning.
</li>
</ol>
</div>
<div class="os-raise-ib-cta-prompt">
<p>Write down your answer, then select the <strong>solution</strong> button to compare your work. </p>
</div>
</div>
<div class="os-raise-ib-content" data-schema-version="1.0" data-wait-for-event="Reveal1">
<p>Compare your answer:</p>
<p>\(f\) is growing faster for the values of \(x\) in the table, but
as \(x\) increases, \(g\) starts to grow faster. So
as \(x\) increases, \(g\) may eventually grow faster than \(f\). </p>
</div>
<br>
<!--Text Entry Interaction Start -->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="c3fb2fe1-8101-48fd-bc49-a77a1483edff"
data-fire-event="eventShow" data-schema-version="1.0">
<div class="os-raise-ib-input-content">
<ol class="os-raise-noindent" start="3">
<li>Which function do you think will reach a value of 2,000 first? Show your reasoning. If you get stuck, consider
increasing \(x\) by 100 a few times and record the function values in the table. </li>
</ol>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>When \(x\) is 600, \(f(600)=1,200\) and \(g(600) \approx 392\), so \(f\) takes a larger value.
When \(x\) is 700, \(f(700)=1,400\) and \(g(700) \approx 1,059\), so \(f\) still takes a larger value.
But when \(x\) is 800, \(f(800)=1,600\) and \(g(800) \approx 2,865\), so \(g\) takes a larger value.
</p>
</div>
</div>
<!--Interaction End -->
<br>
<br>
<!-- "Start "Are you Ready for More? click to reveal. If you have more than one on a page, you will need to change the Data-fire/data-wait-for-events for each set.-->
<div class="os-raise-ib-cta" data-button-text="Are you ready for more?" data-fire-event="RFM1"
data-schema-version="1.0">
<div class="os-raise-ib-cta-content">
<!-- INSERT ANY VALID HTML HERE -->
</div>
<div class="os-raise-ib-cta-prompt">
<!-- INSERT ANY VALID HTML HERE -->
</div>
</div>
<!--Start interaction. If multiple interactions appear under "are you ready for more?, they should all have matching "data-wait-for-event", which should also match the "data-fire-event" for the button. Note in this sample, they are all RFM1-->
<div class="os-raise-ib-input" data-button-text="Solution" data-content-id="dfb7fe6e-d285-441a-9118-a1ecf4d8238c"
data-schema-version="1.0" data-wait-for-event="RFM1">
<div class="os-raise-ib-input-content">
<h4>Extending Your Thinking</h4>
<p>Consider the functions \(g(x)=x^5\) and \(f(x)=5^x\). While it is true that \(f(7)>g(7)\), for example, it is
hard to check this using mental math. Find a value of \(x\) for which properties of exponents allow you to
conclude that \(f(x)>g(x)\) without a calculator.</p>
</div>
<div class="os-raise-ib-input-prompt">
<p>Enter your answer here:</p>
</div>
<div class="os-raise-ib-input-ack">
<p>Compare your answer:</p>
<p>Your answer may vary, but here are some samples: One example is \(x=25\). We have \(f(25)=5^{25}\), whereas
\(g(25)=25^5=(5^2)^5=5^{10}\), which is certainly smaller than \(5^{25}\).</p>
</div>
</div>
<!--End interaction-->