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rational_functions.html
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<h1>Rational functions</h1>
<p>A rational expression is the ratio of two polynomial expressions. Such expressions arise in many modeling situations. As well, as many facts are known about polynomial expressions, much can be determined about rational expressions. This section covers some additional details that arise when graphing such expressions.</p>
<h2>Rational functions</h2>
<p>The rational numbers are simply ratios of integers, of the form <span class="math">$p/q$</span> for non-zero <span class="math">$q$</span>. A rational function is a ratio of <em>polynomial</em> <em>functions</em> of the form <span class="math">$p(x)/q(x)$</span>, again <span class="math">$q$</span> is non-zero, but may have zeros.</p>
<p>We know that polynomials have nice behaviors due to the following facts:</p>
<ul>
<li><p>Behaviors at <span class="math">$-\infty$</span>, <span class="math">$\infty$</span> are known just from the leading term.</p>
</li>
<li><p>There are possible wiggles up and down, the exact behavior depends on intermediate terms, but there can be no more than <span class="math">$n-1$</span> wiggles.</p>
</li>
<li><p>The number of zeros is no more than <span class="math">$n$</span>, the degree of the polynomial.</p>
</li>
</ul>
<p>Rational functions are not quite so nice:</p>
<ul>
<li><p>behavior at <span class="math">$-\infty$</span> and <span class="math">$\infty$</span> can be like a polynomial of any degree, including constants</p>
</li>
<li><p>behaviour at any value x can blow up due to division by 0 – rational functions, unlike polynomials, are not always defined</p>
</li>
<li><p>The function may or may not cross zero, even if the range includes every other point, as the graph of <span class="math">$f(x) =1/x$</span> will show.</p>
</li>
</ul>
<p>Here, as with our discussion on polynomials, we are interested for now in just a few properties:</p>
<ul>
<li><p>What happens to <span class="math">$f(x)$</span> when <span class="math">$x$</span> gets really big or really small (towards <span class="math">$\infty$</span> or <span class="math">$-\infty$</span>)?</p>
</li>
<li><p>What happens near the values where <span class="math">$q(x) = 0$</span>?</p>
</li>
<li><p>When is <span class="math">$f(x) = 0$</span>?</p>
</li>
</ul>
<p>These questions can often be answered with a graph, but with rational functions we will see that care must be take when producing a useful graph.</p>
<p>For example, consider this graph generated from a simple rational function:</p>
<p class="math">\[
~
f(x) = \frac{(x-1)^2 \cdot (x-2)}{(x+3) \cdot (x-3)}.
~
\]</p>
<pre class='hljl'>
<span class='hljl-k'>using</span><span class='hljl-t'> </span><span class='hljl-n'>CalculusWithJulia</span><span class='hljl-t'> </span><span class='hljl-cs'># to load the `Plots` and `SymPy` packages</span><span class='hljl-t'>
</span><span class='hljl-nf'>f</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>=</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-oB'>-</span><span class='hljl-ni'>1</span><span class='hljl-p'>)</span><span class='hljl-oB'>^</span><span class='hljl-ni'>2</span><span class='hljl-t'> </span><span class='hljl-oB'>*</span><span class='hljl-t'> </span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-oB'>-</span><span class='hljl-ni'>2</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-oB'>/</span><span class='hljl-t'> </span><span class='hljl-p'>((</span><span class='hljl-n'>x</span><span class='hljl-oB'>+</span><span class='hljl-ni'>3</span><span class='hljl-p'>)</span><span class='hljl-oB'>*</span><span class='hljl-p'>(</span><span class='hljl-n'>x</span><span class='hljl-oB'>-</span><span class='hljl-ni'>3</span><span class='hljl-p'>)</span><span class='hljl-t'> </span><span class='hljl-p'>)</span><span class='hljl-t'>
</span><span class='hljl-nf'>plot</span><span class='hljl-p'>(</span><span class='hljl-n'>f</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-ni'>10</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>10</span><span class='hljl-p'>)</span>
</pre>
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<p>We would be hard pressed to answer any of the three questions above from the graph, though, on inspection, we might think the strange spikes have something to do with <span class="math">$x$</span> values where <span class="math">$q(x)=0$</span>.</p>
<p>The question of big or small <span class="math">$x$</span> is not answered well with this graph, as the spikes dominate the scale of the <span class="math">$y$</span>-axis. Setting a much larger viewing window illuminates this question:</p>
<pre class='hljl'>
<span class='hljl-nf'>plot</span><span class='hljl-p'>(</span><span class='hljl-n'>f</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-oB'>-</span><span class='hljl-ni'>100</span><span class='hljl-p'>,</span><span class='hljl-t'> </span><span class='hljl-ni'>100</span><span class='hljl-p'>)</span>
</pre>
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