1.6 - Combinations of Functions

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Unformatted text preview: <?xml version="1.0" encoding="utf-8"?> <!DOCTYPE html PUBLIC "-//W3C//DTD XHTML 1.0 Strict//EN" "http://www.w3.org/TR/xhtml1/DTD/xhtml1-strict.dtd"> <html xmlns="http://www.w3.org/1999/xhtml"> <head> <title>1.6 - Combinations of Functions</title> <meta http-equiv="Content-Type" content="text/html; charset=utf-8" /> <link href="../m116.css" rel="stylesheet" type="text/css" /> <style type="text/css"> <!-- .small { font-size: smaller; vertical-align: middle; }--> </style> </head> <body> <h1>1.6 - Combinations of Functions</h1> <h2>Arithmetic Combinations of Functions</h2> <p>The sum, difference, product, or quotient of functions can be found easily. </p> <dl> <dt>Sum</dt> <dd>(f + g)(x) = f(x) + g(x)</dd> <dt>Difference</dt> <dd>(f - g)(x) = f(x) - g(x)</dd> <dt>Product</dt> <dd>(f &middot; g)(x) = f(x) &middot; g(x)</dd> <dt>Quotient</dt> <dd>(f / g)(x) = f(x) / g(x), as long as g(x) isn't zero.</dd> </dl> <p>The domain of each of these combinations is the intersection of the domain of f and the domain of g. In other words, both functions must be defined at a point for the combination to be defined. One additional requirement for the division of functions is that the denominator can't be zero, but we knew that because it's part of the <a href="functions.html">implied domain</a>. </p> <p>Basically what the above says is that to evaluate a combination of functions, you may combine the functions and then evaluate or you may evaluate each function and then combine.</p> <h3>Examples</h3> <p>In the following examples, let f(x) = 5x+2 and g(x) = x<sup>2</sup>-1. We will then evaluate each combination at the point x=4. f(4)=5(4)+2=22 and g(4)=4<sup>2</sup>-1=15 </p> <table border="1" cellpadding="5" cellspacing="0"> <tr class="datath"> <th valign="top">Expression</th> <th colspan="3" valign="top">Combine, then evaluate</th> <th colspan="2" valign="top">Evaluate, then combine</th> </tr> <tr> <td valign="top" class="datal">(f+g)(x)</td> <td valign="top" class="datal">(5x+2) + (x<sup>2</sup>-1)<br /> =x<sup>2</sup>+5x+1</td> <td valign="top" class="datal">(f+g)(4)</td> <td valign="top" class="datal">4<sup>2</sup>+5(4)+1<br /> =16+20+1<br /> =37</td> <td valign="top" class="datal">f(4)+g(4)</td> <td valign="top" class="datal">22+15<br /> =37</td> </tr> <tr> <td valign="top" class="datal">(f-g)(x)</td> <td valign="top" class="datal">(5x+2) - (x<sup>2</sup>-1)<br /> =-x<sup>2</sup>+5x+3</td> <td valign="top" class="datal">(f-g)(4)</td> <td valign="top" class="datal">-4<sup>2</sup>+5(4)+3<br /> =-16+20+3<br /> =7</td> <td valign="top" class="datal">f(4)-g(4)</td> <td valign="top" class="datal">22-15<br /> =7</td></tr> <tr> <td valign="top" class="datal">(f&middot;g)(x)</td> <td valign="top" class="datal">(5x+2)*(x<sup>2</sup>-1)<br /> =5x<sup>3</sup>+2x<sup>2</sup>-5x-2</td> <td valign="top" class="datal">(f&middot;g)(4)</td> <td valign="top" class="datal">5(4<sup>3</sup>)+2(4<sup>2</sup>)-5(4)-2<br /> =5(64)+2(16)-20-2<br /> =330</td> <td valign="top" class="datal">f(4)&middot;g(4)</td>...
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This note was uploaded on 10/18/2011 for the course MAT 1033 taught by Professor Brown during the Spring '10 term at Valencia.

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1.6 Combinations - <?xml version="1.0" encoding="utf-8"><!DOCTYPE html PUBLIC"/W3C/DTD XHTML 1.0

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