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      <text>2.7.2.1 Rational functions</text>
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      <text><![CDATA[<h4>Rational functions</h4>

<p>A rational function is formed by dividing one polynomial by another. Examples include:</p>
<br>
<p style="text-align: center;">\(R_1(x)=\dfrac{x+6}{x^2+1}\), &nbsp;&nbsp;&nbsp;&nbsp; \(R_2(t)=\dfrac{t^3-1}{2t+3}\), &nbsp;&nbsp;&nbsp;&nbsp; \(R_3(z)=\dfrac{2z^2+z-1}{z^2+z-2}\)&nbsp; .</p>
<br>
<p>For convenience we have labelled these rational functions \(R_1\), \(R_2\) and \(R_3\).</p>
<div class="HELM_keypoint">
<h4>Key Point</h4>
<p>A <b>rational function</b> has the form:</p>
<p>\[R(X)=\dfrac{P(x)}{Q(x)}\]</p>
<p>where \(P\) and \(Q\) are polynomial functions and \(Q)\) is nonzero.</p><br>
<p>\(P\) is called the <b>numerator</b> and \(Q\) is called the <b>denominator</b>.</p>
</div>
<p>The graphs of rational functions can take a variety of different forms and can be difficult to plot by hand when you first encounter them. We will see later how to analyse these functions but in the short term the use of a graphics calculator or computer software can help. If you have access to a plotting package or calculator it would be useful to obtain graphs of these functions for yourself. In what follows we explore some of the features of the graphs.</p>
<hr>
<h4 class="HELM_example">Example</h4>
<p>Given the rational function \(R_1(x)=\dfrac{x+2}{x^2+1}\) and its graph in the picture, answer the following questions.</p>
<figure class="HELM_figure">
<img src="@@PLUGINFILE@@/2.7.2.1%20rational%20function1.svg" alt="" role="presentation" class="img-responsive atto_image_button_text-bottom" width="203" height="168"></p><p style="text-align: center;">
</figure>
<ol class="HELM_parts">
<li>For what values of \(x\), if any, is the denominator zero?<br></li>
<li>For what values of \(x\), if any, is the denominator negative?<br></li>
<li>For what values of \(x\) is the function negative?<br></li>
<li>What is the value of the function when \(x\) is zero?<br></li>
<li>What happens to the function as \(x\) gets larger and larger?<br></li>
</ol><p style="text-align: left;"></p>

<h4 class="HELM_solution">Solution</h4>
<ol class="HELM_parts">
<li>\(x^2+ 1\) is never zero.</li>
<li>\(x^2+ 1\) is never negative, it is always positive.</li>
<li>Only when the numerator \(x+ 2\) is negative which is when \(x\) is less than \(−2\).</li>
<li>\(2\), because that is when the numerator \(x+ 2 = 0\).</li>
<li> \(R_1\) approaches zero because the \(x^2\) term in the denominator becomes very large. (This is seen by substituting larger and larger values e.g. \(10,100,1000...\)).</li>
</ol><p style="text-align: left;"></p>
<hr>

<p>Note that for large \(x\) values the graph gets closer and closer to the \(x\)-axis. We say that the \(x\)-axis is a <strong>horizontal asymptote</strong> of this graph.</p>
<p>Answering questions such as (a) to (c) above will help you to sketch graphs of rational functions.<p>]]></text>
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