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156 lines
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HTML
156 lines
7.1 KiB
HTML
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<!-- #BeginEditable "Body" -->
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<h1 class="center">Even and Odd Functions</h1>
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<p class="center">They are special types of functions</p>
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<h2>Even Functions</h2>
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<p>A function is "even" when:</p>
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<p class="center"><span class="larger">f(x) = f(−x) </span>for all x</p>
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<p>In other words there is <a href="equation-symmetry.html">symmetry about the y-axis</a> (like a reflection):</p>
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<p class="center"><img src="images/function-even.svg" alt="Even Function" /></p>
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<p class="center larger">This is the curve f(x) = x<sup>2</sup>+1</p>
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<p>They got called "even" functions because the functions x<sup>2</sup>, x<sup>4</sup>, x<sup>6</sup>, x<sup>8</sup>, etc behave like that, but there are other functions that behave like that too, such as cos(x):</p>
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<p class="center"><img src="images/function-even-cos.gif" alt="cos(x)" width="464" height="105" /><br>
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<span class="larger">Cosine function: f(x) = cos(x)</span><br>
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It is an even function
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<br></p>
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<p>But an even exponent does not always make an even function, for example <span class="larger">(x+1)<sup>2</sup></span> is <b>not</b> an even function.</p>
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<p> </p>
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<h2>Odd Functions</h2>
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<p>A function is "odd" when:</p>
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<p class="center larger">−f(x) = f(−x) for all x</p>
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<p>Note the minus in front of f(x): <b>−f(x)</b>.</p>
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<p>And we get <a href="../geometry/symmetry-point.html">origin symmetry</a>:</p>
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<p class="center"><img src="images/function-odd.svg" alt="Odd Function" /></p>
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<p class="center"><span class="larger">This is the curve f(x) = x<sup>3</sup>−x</span></p>
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<p>They got called "odd" because the functions x, x<sup>3</sup>, x<sup>5</sup>, x<sup>7</sup>, etc behave like that, but there are other functions that behave like that, too, such as<b> sin(x)</b>:</p>
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<p class="center"><img src="images/function-odd-sin.gif" alt="sin(x)" width="408" height="94" /><br>
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<span class="larger">Sine function: f(x) = sin(x)</span><br>
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It is an odd function</p>
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<p>But an odd exponent does not always make an odd function, for example <span class="larger">x<sup>3</sup>+1</span> is <b>not</b> an odd function.</p>
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<h2>Neither Odd nor Even</h2>
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<p>Don't be misled by the names "odd" and "even" ... they are just <b>names</b> ... and a function does <b>not have to be</b> even or odd.</p>
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<p>In fact most functions are neither odd nor even. For example, just adding 1 to the curve above gets this:<span class="larger"></span></p>
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<p class="center"><img src="images/function-not-even-odd.svg" alt="Not Even Nor Odd Function" /></p>
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<p class="center"><span class="larger">This is the curve f(x) = x<sup>3</sup>−x<b>+1</b></span></p>
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<p class="center">It is <b>not an odd function</b>, and it is <b>not an even function</b> either.<br>
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It is neither odd nor even</p>
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<h2>Even or Odd?</h2>
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<div class="example">
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<h3>Example: is f(x) = x/(x<sup>2</sup>−1) Even or Odd or neither?</h3>
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<p>Let's see what happens when we substitute <b>−x</b>:</p>
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<div class="tbl">
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<div class="row"><span class="left"><span class="larger"><b>f(−x)</b> =</span></span><span class="right"> (−x)/((−x)<sup>2</sup>−1)</span></div>
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<div class="row"><span class="left">=</span><span class="right">−x/(x<sup>2</sup>−1) </span></div>
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<div class="row"><span class="left">=</span><span class="right"><b>−f(x)</b></span></div>
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</div>
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<p> </p>
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<p class="center">So <span class="large">f(−x) = −f(x)</span> , which makes it an <b>Odd Function</b></p>
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</div>
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<h2>Even and Odd</h2>
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<p>The only function that is even <b>and</b> odd is f(x) = 0</p>
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<h2>Special Properties</h2>
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<p>Adding:</p>
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<ul>
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<li>The sum of two even functions is even</li>
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<li>The sum of two odd functions is odd</li>
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<li>The sum of an even and odd function is neither even nor odd (unless one function is zero).</li>
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</ul>
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<p>Multiplying:</p>
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<ul>
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<li>The product of two even functions is an even function.</li>
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<li>The product of two odd functions is an even function.</li>
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<li>The product of an even function and an odd function is an odd function.</li>
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</ul>
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<p> </p>
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<div class="questions">
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<script>getQ(547, 548, 549, 550, 1184, 1185, 1186, 8454, 8455, 8456);</script>
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</div>
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<div class="related">
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<a href="equation-symmetry.html">Equation Symmetry</a>
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<a href="../sets/function-hyperbolic.html">Hyperbolic Functions</a>
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<a href="index.html">Algebra Index</a>
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