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<title>Composition of Functions</title>
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<h1 align="center">Composition of Functions</h1>
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<p>"Function Composition" is applying one function to the results of another:</p>
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<p align="center"><img src="images/function-composition.svg" alt="Function Composition" /></p>
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<p align="center">The result of <span class="large">f()</span> is sent through <span class="large">g()</span></p>
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<p align="center">It is written: <span class="largest">(g <span style="font-size: 75%;">º</span> f)(x)</span></p>
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<p align="center">Which means: <span class="largest">g(f(x))</span></p>
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<p> </p>
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<div class="example">
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<h3>Example: <b>f(x) = 2x+3</b> and <b>g(x) = x<sup>2</sup></b></h3>
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<p><b>"x" is just a placeholder</b>. To avoid confusion let's just call it "input":</p>
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<p align="center"><b>f(input) = 2(input)+3</b></p>
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<p align="center"><b>g(input) = (input)<sup>2</sup></b></p>
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<p>Let's start:</p>
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<p align="center" class="large">(g <span style="font-size: 75%;">º</span> f)(x) = g(f(x))</p>
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<p>First we apply <span class="large">f</span>, then apply <span class="large">g</span> to that result:</p>
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<p align="center"><img src="images/function-composition-a.svg" alt="Function Composition" /></p>
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<p align="center" class="large">(g <span style="font-size: 75%;">º</span> f)(x) = (2x+3)<sup>2</sup></p>
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<p> </p>
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<p>What if we <b>reverse</b> the order of <span class="large">f</span> and <span class="large">g</span>?</p>
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<p align="center" class="large">(f <span style="font-size: 75%;">º</span> g)(x) = f(g(x))</p>
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<p>First we apply <span class="large">g</span>, then apply <span class="large">f</span> to that result:</p>
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<p align="center"><img src="images/function-composition-b.svg" alt="Function Composition" /></p>
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<p align="center" class="large">(f <span style="font-size: 75%;">º</span> g)(x) = 2x<sup>2</sup>+3</p>
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<p align="center"> </p>
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<p><b>We get a different result!</b> </p>
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<p>When we reverse the order the result is rarely the same.</p>
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<p>So be careful which function comes first.</p>
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</div>
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<h2>Symbol</h2>
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<div class="center80">
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<p>The symbol for composition is a small circle:</p>
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<p class="center largest">(g <span style="font-size: 75%;">º</span> f)(x)</p>
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</div>
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<p>It is <b>not</b> a filled in dot: <span class="large">(g · f)(x)</span>, as that means <b>multiply</b>.</p>
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<h2>Composed With Itself</h2>
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<p>We can even compose a function with itself!</p>
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<div class="example">
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<h3>Example: <b>f(x) = 2x+3</b></h3>
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<p> </p>
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<p align="center" class="large">(f <span style="font-size: 75%;">º</span> f)(x) = f(f(x))</p>
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<p>First we apply <span class="large">f</span>, then apply <span class="large">f</span> to that result:</p>
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<p align="center"><img src="images/function-composition-d.svg" alt="Function Composition" /></p>
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<p align="center" class="large">(f <span style="font-size: 75%;">º</span> f)(x) = 2(2x+3)+3 = 4x + 9</p>
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<p>We should be able to do it without the pretty diagram:</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="large">(f <span style="font-size: 75%;">º</span> f)(x)</span></span><span class="right">= <span class="large"> f(f(x))</span></span></div>
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<div class="row"><span class="left"> </span><span class="right">= <span class="large"> f(2x+3)</span></span></div>
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<div class="row"><span class="left"> </span><span class="right">= <span class="large">2(2x+3)+3</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="large">= 4x + 9</span></span></div>
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</div>
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</div>
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<h2>Domains</h2>
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<p>It has been easy so far, but now we must consider the <b><a href="domain-range-codomain.html">Domains</a></b> 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 graph" /></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), so we must <b>exclude</b> negative numbers:</p>
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<p align="center" class="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 align="center"><img src="images/interval-0-on.gif" alt="zero onwards" width="216" height="50" /></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 align="center"><span class="larger">{ x<img src="../images/symbols/member-of-sm.gif" alt="member of" width="14" height="19" style="vertical-align:middle;" /><img src="../images/symbols/set-r.svg" alt="reals" height="16" style="vertical-align:middle;" /> | x ≥ 0}</span></p>
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<p>Or using <a href="intervals.html">interval notation</a> it is:</p>
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<p align="center" class="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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<h2>Domain of Composite Function</h2>
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<p>We must get <b>both Domains</b> right (the composed function <b>and</b> the first function used).</p>
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<p>When doing, for example, <span class="large">(g <span style="font-size: 75%;">º</span> f)(x) = g(f(x))</span>:</p>
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<ul>
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<li>Make sure we get the Domain for <b>f(x)</b> right,</li>
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<li>Then also make sure that <b>g(x)</b> gets the correct Domain</li>
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</ul>
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<div class="example">
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<h3>Example: <b>f(x) = √x</b> and <b>g(x) = x<sup>2</sup></b></h3>
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<p class="larger">The Domain of <b>f(x) = √x</b> is all non-negative Real Numbers</p>
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<p class="larger">The Domain of <b>g(x) = x<sup>2</sup></b> is all the Real Numbers</p>
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<p>The composed function is:</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="large">(g <span style="font-size: 75%;">º</span> f)(x) </span></span><span class="right"><span class="large">= g(f(x)) </span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="large">= (√x)<sup>2</sup></span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="large">= x</span></span></div>
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</div>
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<p>Now, "x" normally has the Domain of all Real Numbers ...</p>
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<p align="center">... but because it is a <b>composed function</b> we must<b> also consider f(x)</b>,</p>
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<p class="larger">So the Domain is all non-negative Real Numbers</p>
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</div>
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<h2>Why Both Domains?</h2>
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<p>Well, imagine the functions are machines ... the first one melts a hole with a flame (only for metal), the second one drills the hole a little bigger (works on wood or metal):</p>
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<p align="center"><img src="images/function-composition-c.svg" alt="Function Composition" /></p>
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<table width="85%" border="0" align="center">
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<tr>
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<td><img src="../images/flame.jpg" alt="Fire" width="90" height="120" /></td>
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<td><p>What we see at the end is a drilled hole, and we may think "that should work for wood <b>or</b> metal". </p>
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<p>But if we put wood into <span class="large">g <span style="font-size: 75%;">º</span> f</span> then the first function <span class="large">f</span> will make a fire and burn everything down!</p></td>
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</tr>
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</table>
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<p align="center" class="larger">So what happens "inside the machine" is important.</p>
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<p> </p>
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<h2>De-Composing Function</h2>
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<p>We can go the other way and <b>break up a function</b> into a composition of other functions.</p>
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<div class="example">
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<h3>Example: <b> (x+1/x)<sup>2</sup></b></h3>
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<p>That function can be made from these two functions:</p>
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<p align="center" class="large">f(x) = x + 1/x</p>
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<p align="center" class="large">g(x) = x<sup>2</sup></p>
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<p>And we get:</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="large">(g <span style="font-size: 75%;">º</span> f)(x) </span></span><span class="right"><span class="large">= g(f(x))</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="large">= g(x + 1/x)</span></span></div>
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<div class="row"><span class="left"> </span><span class="right"><span class="large">= (x + 1/x)<sup>2</sup></span></span></div>
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</div>
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</div>
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<p>This can be useful if the original function is too complicated to work on.</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>"Function Composition" is applying one function to the results of another.</li>
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<li><b>(g <span style="font-size: 75%;">º</span> f)(x) = g(f(x))</b>, first apply f(), then apply g()</li>
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<li>We must also respect the domain of the first function</li>
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<li>Some functions can be de-composed into two (or more) simpler functions.</li>
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</div>
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</ul>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(561,562,2427, 2428, 563,564,1195,1196,2429, 2430);</script>
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</div>
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<div class="related">
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<a href="functions-operations.html">Operations with Functions</a>
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<a href="../algebra/index.html">Algebra Index</a> </div>
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