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286 lines
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HTML
286 lines
12 KiB
HTML
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<html lang="en"><!-- #BeginTemplate "/Templates/Advanced.dwt" --><!-- DW6 -->
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<!-- Mirrored from www.mathsisfun.com/sets/functions-operations.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:03:18 GMT -->
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<meta http-equiv="content-type" content="text/html; charset=UTF-8">
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<title>Operations with Functions</title>
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<div id="hdr"></div>
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<div id="cookOK"></div>
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<article id="content" role="main">
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Operations with Functions</h1>
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<table align="center" width="80%" border="0">
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<tbody>
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<tr>
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<td>
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<img src="../images/style/plus.svg" alt="plus" height="" width="">
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<img src="../images/style/minus.svg" alt="minus" height="" width="">
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<br>
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<img src="../images/style/multiply.svg" alt="multiply" height="" width="">
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<img src="../images/style/divide.svg" alt="divide" height="" width="">
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</td>
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<td>
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<p>We can add, subtract, multiply and divide functions!</p>
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<p>The result is a new function.</p></td>
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</tr>
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</tbody></table>
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<p>Let us try doing those operations on <span class="largest">f(x) </span>and <span class="largest">g(x)</span>:</p>
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<table style="border: 0;">
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<tbody>
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<tr>
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<td style="text-align:right;"><img src="../images/style/plus.svg" alt="add" height="" width=""></td>
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<td>
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<h3>Addition</h3></td>
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</tr></tbody></table>
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<p>We can add two functions: </p>
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<p class="largest" align="center">(f+g)(x) = f(x) + g(x)</p>
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<p class="center"><i>Note: we put the <b>f+g</b> inside <b>()</b> to show they both work on <b>x</b>.</i></p>
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<div class="example">
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<h3>Example: f(x) = 2x+3 and g(x) = x<sup>2</sup></h3>
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<p class="center large">(f+g)(x) = (2x+3) + (x<sup>2</sup>) = x<sup>2</sup>+2x+3</p>
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</div>
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<p>Sometimes we may need to <a href="../algebra/like-terms.html">combine like terms</a>:</p>
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<div class="example">
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<h3>Example: v(x) = 5x+1, w(x) = 3x-2</h3>
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<p class="center large">(v+w)(x) = (5x+1) + (3x-2) = 8x-1</p>
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</div>
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<p>The only other thing to worry about is the Domain (the set of numbers that go into the function), but we will talk about that later!</p>
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<p> </p>
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<table>
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<tbody>
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<tr>
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<td style="text-align:right;"><img src="../images/style/minus.svg" alt="Subtract" height="40" width="40"></td>
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<td>
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<h3>Subtraction</h3></td>
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</tr></tbody></table>
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<p>We can also subtract two functions:</p>
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<p class="largest" align="center">(f-g)(x) = f(x) − g(x)</p>
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<div class="example">
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<h3>Example: f(x) = 2x+3 and g(x) = x<sup>2</sup></h3>
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<p class="center large">(f-g)(x) = (2x+3) − (x<sup>2</sup>)</p>
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</div>
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<p> </p>
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<table>
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<tbody>
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<tr>
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<td style="text-align:right;"><img src="../images/style/multiply.svg" alt="Multiply" height="40" width="40"></td>
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<td>
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<h3>Multiplication</h3></td>
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</tr></tbody></table>
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<p>We can multiply two functions:</p>
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<p class="largest" align="center">(f·g)(x) = f(x) · g(x)</p>
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<div class="example">
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<h3>Example: f(x) = 2x+3 and g(x) = x<sup>2</sup></h3>
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<p class="center large">(f<span class="largest">·</span>g)(x) = (2x+3)(x<sup>2</sup>) = 2x<sup>3</sup> + 3x<sup>2</sup></p>
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</div>
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<p> </p>
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<table>
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<tbody>
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<tr>
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<td style="text-align:right;"><img src="../images/style/divide.svg" alt="Divide" height="40" width="40"></td>
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<td>
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<h3>Division</h3></td>
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</tr>
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</tbody></table>
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<p>And we can divide two functions:</p>
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<p class="largest" align="center">(f/g)(x) = f(x) / g(x)</p>
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<div class="example">
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<h3>Example: f(x) = 2x+3 and g(x) = x<sup>2</sup></h3>
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<p class="center large">(f/g)(x) = (2x+3)/x<sup>2</sup></p>
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</div>
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<h2>Function Composition</h2>
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<table style="border: 0; margin:auto;">
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<tbody>
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<tr>
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<td>There is another special operation called <a href="functions-composition.html">Function Composition</a>,<br>
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read that page to find out more!</td>
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<td> </td>
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<td align="center" nowrap="nowrap"><span class="largest">(g <span style="font-size: 75%;">º</span> f)(x)</span></td>
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</tr>
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</tbody></table>
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<h2>Domains</h2>
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<p>It has been easy so far, but now we must consider the <a href="domain-range-codomain.html">Domains</a> of the functions.</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/range-domain-graph.svg" alt="domain and range on a graph" height="167" width="298"></p>
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<p>The domain is <b>the set of all the values</b> that go into a function.</p>
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<p>The function must work for all values we give it, so it is <b>up to us</b> to make sure we get the domain correct!</p>
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<div style="clear:both"></div>
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<div class="example">
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<h3>Example: the domain for √x (the square root of x)</h3>
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<p>We can't have the square root of a negative number (unless we use imaginary numbers, but we aren't doing that here), so we must <b>exclude</b> negative numbers:</p>
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<p class="center larger">The Domain of √x is all non-negative Real Numbers</p>
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<p>On the Number Line it looks like:</p>
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<p class="center"><img src="images/interval-0-on.svg" alt="zero onwards" height="" width=""></p>
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<p>Using <a href="set-builder-notation.html">set-builder notation</a> it is written:</p>
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<p class="center"><span class="larger">{ x<img src="../images/symbols/member-of.svg" alt="member of" style="vertical-align:middle;" height="" width=""><span class="large"><img src="../images/symbols/set-r.svg" alt="Reals" style="vertical-align:middle;" height="23" width="23"></span> | x ≥ 0}</span></p>
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<p class="center"><i>"the set of all x's that are a member of the Real Numbers,<br>
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such that x is greater than or equal to zero</i>"</p>
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<p>Or using <a href="intervals.html">interval notation</a> it is:</p>
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<p class="center larger">[0,+∞)</p>
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</div>
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<p>It is important to get the Domain right, or we will get bad results!</p>
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<p>So how do we work out the new domain after doing an operation?</p>
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<h2>How to Work Out the New Domain</h2>
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<p>When we do operations on functions, we end up with the <b>restrictions of both</b>.</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="images/chicken-dish.jpg" alt="chicken dish" height="130" width="172"></p>
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<p>It is like cooking for friends:</p>
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<ul>
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<li>one can't eat peanuts,</li>
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<li>the other can't eat dairy food.</li>
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</ul>
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<p>So what we cook can't have peanuts <b>and also</b> can't have dairy products.</p>
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<div class="example">
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<h3>Example: f(x)=√x and g(x)=√(3−x)</h3>
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<p>The domain for <b>f(x)=√x</b> is from 0 onwards:</p>
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<p class="center"><img src="images/interval-0-on.svg" alt="zero onwards" height="" width=""></p>
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<p>The domain for <b>g(x)=√(3−x)</b> is up to and including 3:</p>
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<p class="center"><img src="images/interval-to-3.svg" alt="zero onwards" height="" width=""></p>
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<p>So the new domain (after adding or whatever) is from 0 to 3:</p>
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<p class="center"><img src="images/interval-0-3.svg" alt="zero onwards" height="" width=""></p>
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<p>If we choose any other value, then one or the other part of the new function won't work.</p>
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</div>
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<p>In other words we want to find where the two domains <b>intersect</b>.</p>
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<div class="center80">
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<p>Note: we can put this whole idea into one line using <a href="set-builder-notation.html">Set Builder Notation</a>:</p>
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<p class="center larger">Dom(f+g) = { x<span class="larger"><img src="../images/symbols/member-of.svg" alt="member of" style="vertical-align:middle;" height="" width=""></span><span class="large"><img src="../images/symbols/set-r.svg" alt="Reals" style="vertical-align:middle;" height="23" width="23"></span> | x<span class="larger"><img src="../images/symbols/member-of.svg" alt="member of" style="vertical-align:middle;" height="" width=""></span>Dom(f) and x<span class="larger"><img src="../images/symbols/member-of.svg" alt="member of" style="vertical-align:middle;" height="" width=""></span>Dom(g) }</p>
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<p>Which says "the domain of f plus g is the set of all Real Numbers that are in the domain of f AND in the domain of g" (<img src="../images/symbols/member-of.svg" alt="member of" style="vertical-align:middle;" height="" width="">means "member of")</p>
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<p>The same rule applies when we add, subtract, multiply or divide, except divide has one extra rule.</p>
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</div>
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<h2>An Extra Rule for Division</h2>
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<p>There is an <b>extra rule</b> for division:</p>
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<p class="center"><b>As well as</b> restricting the domain as above, when we <b>divide</b>:</p>
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<p class="center"><span class="largest">(f/g)(x) = f(x) / g(x)</span></p>
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<p class="center"><b>we must also</b> make sure that g(x) is <b>not equal to zero</b> (so we don't <a href="../numbers/dividing-by-zero.html">divide by zero</a>).</p>
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<p>Here is an example:</p>
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<div class="example">
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<h3>Example: f(x)=√x and g(x)=√(3−x)</h3>
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<p class="largest" align="center">(f/g)(x) = √x / √(3−x)</p>
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<p>1. The domain for <b>f(x)=√x</b> is from 0 onwards:</p>
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<p class="center"><img src="images/interval-0-on.svg" alt="zero onwards" height="" width=""></p>
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<p>2. The domain for <b>g(x)=√(3−x)</b> is up to and including 3:</p>
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<p class="center"><img src="images/interval-to-3.svg" alt="zero onwards" height="" width=""></p>
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<p>3. AND <b>√(3−x) cannot be zero</b>, so x cannot be 3:</p>
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<p class="center"><img src="images/interval-not-3.svg" alt="zero onwards" height="" width=""><br>
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(Notice the <b>open circle</b> at 3, which means <b>not including</b> 3)</p>
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<p>So all together we end up with:</p>
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<p class="center"><img src="images/interval-0-not3.svg" alt="zero onwards" height="" width=""></p></div>
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<p> </p>
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<h2>Summary</h2>
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<ul>
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<div class="bigul">
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<li>To add, subtract, multiply or divide functions just do as the operation says.</li>
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<li>The domain of the new function will have the restrictions of both functions that made it.</li>
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<li>Divide has the extra rule that the function we are dividing by cannot be zero.</li>
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</div>
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</ul>
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<p> </p>
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<div class="questions">1193, 1194, 2459, 2460, 8461, 559, 560, 2461, 2462, 8462</div>
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<div class="related">
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<a href="functions-composition.html">Function Composition</a>
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<a href="../algebra/index.html">Algebra Index</a>
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</div>
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<!-- #EndEditable -->
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</article>
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<div id="adend" class="centerfull noprint"></div>
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<footer id="footer" class="centerfull noprint"></footer>
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<div id="copyrt">Copyright © 2022 Rod Pierce</div>
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