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143 lines
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143 lines
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<title>Exponential Function Reference</title>
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<h1 align="center">Exponential Function Reference</h1>
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<p>This is the general Exponential Function (see below for e<sup>x</sup>):</p>
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<p align="center" class="largest">f(x) = a<sup>x</sup></p>
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<p align="center"><b>a</b> is any value greater than 0</p>
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<h3>Properties depend on value of "a" </h3>
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<ul>
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<div class="bigul">
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<li>When <b>a=1</b>, the graph is a horizontal line at <b>y=1</b></li>
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<li>Apart from that there are two cases to look at:</li>
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</ul>
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<p> </p>
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<div style="float:left; width: 320px; padding: 5px; border: 1px solid blue;">
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<p class="larger center"><b>a</b> between 0 and 1</p>
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<p class="center"><img src="images/function-exponential-lt1.gif" alt="exponential function" width="291" height="169" /><span class="larger"><br>
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Example: <b>f(x) = (0.5)<sup>x</sup></b></span></p>
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<p>For <b>a</b> between 0 and 1</p>
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<ul>
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<li>As <b>x</b> increases, <b>f(x)</b> heads to 0</li>
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<li>As <b>x</b> decreases, <b>f(x)</b> heads to infinity</li>
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<li>It is a <a href="functions-increasing.html">Strictly Decreasing</a> function (and so is "Injective")</li>
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<li>It has a <a href="../algebra/asymptote.html">Horizontal Asymptote</a> along the x-axis (y=0). </li>
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</ul>
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</div>
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<div style="float:right; width: 320px; padding: 5px; border: 1px solid blue;">
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<p class="larger center"><b>a</b> above 1</p>
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<p class="center"><img src="images/function-exponential-gt1.gif" alt="exponential function" width="293" height="169" /><span class="larger"><br>
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Example: <b>f(x) = (2)<sup>x</sup></b></span></p>
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<p>For <b>a</b> above 1:</p>
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<ul>
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<li>As <b>x</b> increases, <b>f(x)</b> heads to infinity</li>
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<li>As <b>x</b> decreases, <b>f(x)</b> heads to 0</li>
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<li>it is a <a href="functions-increasing.html">Strictly Increasing</a> function (and so is "Injective")</li>
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<li>It has a <a href="../algebra/asymptote.html">Horizontal Asymptote</a> along the x-axis (y=0). </li>
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</ul>
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</div><div style="clear:both"></div>
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<p align="center">Plot the graph <a href="../data/function-grapherd8d9.html?func1=a^x&xmin=-12&xmax=12&ymin=-8&ymax=8">here</a> (use the "a" slider)</p>
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<h3>In General:</h3>
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<ul>
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<div class="bigul">
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<li>It is <b>always greater than 0</b>, and never crosses the x-axis</li>
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<li>It always intersects the y-axis at y=1 ... in other words it passes through <b>(0,1)</b></li>
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<li>At <b>x=1</b>, <b>f(x)=a</b> ... in other words it passes through <b>(1,a)</b></li>
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<li>It is an <a href="injective-surjective-bijective.html">Injective (one-to-one)</a> function</li>
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</div>
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</ul>
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<div class="center80">
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<p>Its Domain is the <a href="../numbers/real-numbers.html">Real Numbers</a>: <img src="../images/symbols/set-r.svg" alt="Real Numbers" style="vertical-align:middle;" /></p>
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<p>Its Range is the Positive Real Numbers: <span class="large">(0, +∞)</span></p>
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</div>
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<h2>Inverse</h2>
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<div class="def">
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<p align="center"><span class="large">a<sup>x</sup></span> is the <a href="function-inverse.html">inverse function</a> of <span class="large">log<sub>a</sub>(x)</span> (the <a href="function-logarithmic.html">Logarithmic Function</a>)</p>
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</div>
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<p>So the Exponential Function can be "reversed" by the Logarithmic Function.</p>
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<h2>The Natural Exponential Function</h2>
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<p>This is the <b>"Natural</b>" Exponential Function:</p>
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<p align="center" class="largest">f(x) = e<sup>x</sup></p>
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<p align="center">Where <b>e</b> is "<a href="../numbers/e-eulers-number.html">Eulers Number</a>" = <b>2.718281828459... etc</b></p>
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<p align="center"><img src="images/function-exponential-e.svg" alt="natural exponential function" /><br />
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<span class="larger">Graph of <b>f(x) = e<sup>x</sup></b></span></p>
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<p>The value <i><b>e</b></i> is important because it creates these useful properties:</p>
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<div class="def">
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<p>At any point the slope of <i><b>e</b></i><b></b><sup>x</sup> equals the value of <i><b>e</b></i><b></b><sup>x</sup> : </p>
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<p align="center"><img src="images/function-exponential-slopes.svg" alt="natural exponential function" /><br>
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when x=0, the value of <i><b>e</b></i><sup>x</sup> = <i><b>1</b></i>, and <i><b>slope = 1</b></i><br>
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when x=1, the value of <i><b>e</b></i><sup>x</sup> = <i><b>e</b></i>, and <i><b>slope = e</b></i><br>
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etc...</p>
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<p>The area <b>up to</b> any x-value is also equal to <b><i>e</i></b><sup>x</sup> :</p>
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<p align="center"><img src="images/function-exponential-area.svg" alt="natural exponential function" /></div></p>
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<p> </p>
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<div class="related"><a href="../algebra/exponents-logarithms.html">Exponents and Logarithms</a><a href="functions-common.html">Common Functions Reference</a> <a href="../algebra/index.html">Algebra Index</a></div>
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