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<!-- Mirrored from www.mathsisfun.com/sets/functions-increasing.html by HTTrack Website Copier/3.x [XR&CO'2014], Sat, 29 Oct 2022 01:03:15 GMT -->
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<!-- #BeginEditable "doctitle" -->
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<title>Increasing and Decreasing Functions</title>
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<div id="content" role="main"><!-- #BeginEditable "Body" -->
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<h1 align="center">Increasing and Decreasing Functions</h1>
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<h2>Increasing Functions</h2>
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<p>A <a href="function.html">function</a> is "increasing" when the <b>y-value</b> increases as the <b>x-value</b> increases, like this:</p>
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<p align="center"><img src="../algebra/images/function-increasing.gif" alt="Increasing Function" width="284" height="231" /></p>
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<p align="center">It is easy to see that <b>y=f(x)</b> tends to go <b>up</b> as it goes <b>along</b>.</p>
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<h3>Flat?</h3>
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<p>What about that flat bit near the start? Is that OK?</p>
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<ul>
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<li>Yes, it is OK when we say the function is <b>Increasing</b></li>
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<li>But it is <b>not OK</b> if we say the function is <b>Strictly Increasing</b> (no flatness allowed)</li>
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</ul>
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<h3>Using Algebra</h3>
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<p>What if we can't plot the graph to see if it is increasing? In that case we need a definition using algebra.</p>
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<p>For a function <b>y=f(x)</b>:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td align="center"><span class="larger">when x<sub>1</sub> < x<sub>2</sub> then f(x<sub>1</sub>) ≤ f(x<sub>2</sub>) </span></td>
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<td> </td>
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<td><b>Increasing</b></td>
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</tr>
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<tr>
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<td align="center"><span class="larger">when x<sub>1</sub> < x<sub>2</sub> then f(x<sub>1</sub>) < f(x<sub>2</sub>) </span></td>
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<td> </td>
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<td><b>Strictly Increasing</b></td>
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</tr>
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</table>
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</div>
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<p>That has to be true for <b>any</b> x<sub>1</sub>, x<sub>2</sub>, not just some nice ones we might choose.</p>
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<div class="center80">
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<p align="center">The important parts are <a href="../equal-less-greater.html">the <span class="larger"><</span> and <span class="larger">≤</span> signs</a> ... remember where they go!</p>
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</div><p> </p>
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<h3>An Example:</h3>
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<table border="0" align="center">
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<tr align="center">
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<td><img src="../algebra/images/function-increasing-ex.gif" alt="Increasing Function" width="218" height="180" /></td>
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</tr>
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<tr align="center">
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<td>This is also an increasing function<br />
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even though the rate of increase reduces</td>
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</tr>
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</table>
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<h3>For An Interval</h3>
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<p>Usually we are only interested in <b> some interval</b>, like this one:</p>
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<p align="center"><img src="../algebra/images/function-increasing-interval.gif" alt="Increasing Function" width="220" height="220" /></p>
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<p align="center">This function is <b>increasing</b> for the interval shown<br />
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(it may be increasing or decreasing elsewhere)</p>
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<h2>Decreasing Functions</h2>
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<p>The <b>y-value</b> <span class="larger">decreases</span> as the <b>x-value</b> increases:</p>
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<p align="center"><img src="../algebra/images/function-decreasing2.gif" alt="Decreasing Function" width="283" height="229" /></p>
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<p>For a function <b>y=f(x)</b>:</p>
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<div class="simple">
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<table border="0" align="center">
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<tr>
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<td align="center"><span class="larger">when x<sub>1</sub> < x<sub>2</sub> then f(x<sub>1</sub>) ≥ f(x<sub>2</sub>) </span></td>
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<td> </td>
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<td><b>Decreasing</b></td>
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</tr>
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<tr>
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<td align="center"><span class="larger">when x<sub>1</sub> < x<sub>2</sub> then f(x<sub>1</sub>) > f(x<sub>2</sub>) </span></td>
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<td> </td>
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<td><b>Strictly Decreasing</b></td>
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</tr>
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</table>
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</div>
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<p>Notice that f(x<sub>1</sub>) is now larger than (or equal to) f(x<sub>2</sub>).</p>
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<h2>An Example</h2>
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<p>Let us try to find where a function is increasing or decreasing.</p>
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<p align="center"></p>
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<div class="example">
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<h3>Example: f(x) = x<sup>3</sup>−4x, for x in the interval [−1,2]</h3>
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<p>Let us plot it, including the interval [−1,2]:</p>
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<p style="float:left; margin: 0 10px 5px 0;"><img src="../algebra/images/function-increasing-ex2.gif" alt="Example Function" width="249" height="173" /></p>
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<p>Starting from −1 (the beginning of the interval <b>[−1,2]</b>):</p>
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<ul>
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<li>at x = <b>−1</b> the function is decreasing, </li>
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<li>it continues to decrease until <b>about 1.2</b></li>
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<li>it then increases from there, past x = <b>2</b></li>
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</ul>
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<p>Without exact analysis we cannot pinpoint where the curve turns from decreasing to increasing, so let us just say:</p>
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<p>Within the interval <b>[−1,2]</b>:</p>
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<ul>
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<li>the curve decreases in the interval<b> [−1, approx 1.2]</b></li>
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<li>the curve increases in the interval <b>[approx 1.2, 2]</b></li>
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</ul>
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</div>
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<h2>Constant Functions</h2>
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<p>A Constant Function is a horizontal line:</p>
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<p align="center"><img src="../algebra/images/function-constant.gif" alt="Constant Function" width="218" height="196" /></p>
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<h2>Lines</h2>
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<p>In fact lines are either increasing, decreasing, or constant.</p>
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<p>The <a href="../equation_of_line.html">equation of a line</a> is:</p>
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<p align="center" class="larger">y = mx + b</p>
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<p style="float:right; margin: 0 0 5px 10px;"><img src="../algebra/images/function-increasing-lines.svg" alt="Constant Function" />
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</p> <p>The slope <span class="large">m</span> tells us if the function is increasing, decreasing or constant:</p>
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<table border="0" align="center">
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<tr align="center" class="larger">
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<td class="larger">m < 0</td>
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<td width="20" class="larger"> </td>
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<td class="larger">decreasing</td>
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</tr>
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<tr align="center" class="larger">
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<td class="larger">m = 0</td>
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<td class="larger"> </td>
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<td class="larger">constant</td>
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</tr>
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<tr align="center" class="larger">
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<td class="larger">m > 0</td>
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<td class="larger"> </td>
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<td class="larger">increasing</td>
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</tr>
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</table>
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<div style="clear:both"></div>
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<h2>One-to-One</h2>
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<p>Strictly Increasing (and Strictly Decreasing) functions have a special property called "injective" or "one-to-one" which simply means we never get the same "y" value twice.</p>
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<div style="text-align:center;">
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<p style="display:inline-block; text-align:center; margin:9px;"><img src="images/function-general-graph.svg" alt="General Function" /><br>
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<span class="larger">General Function</span></p>
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<p style="display:inline-block; text-align:center; margin:9px;"><img src="images/function-injective-graph.svg" alt="Injective Function" /><br>
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<span class="larger">"Injective" (one-to-one)</span></p>
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</div>
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<div class="center80">
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<p>Why is this useful? Because Injective Functions can be <b>reversed</b>! </p>
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<p>We can go from a "y" value <b>back to</b> an "x" value (which we can't do when there is more than one possible "x" value).</p>
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</div>
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<p>Read <a href="injective-surjective-bijective.html">Injective, Surjective and Bijective</a> to find out more.</p>
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<p> </p>
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<div class="questions">
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<script type="text/javascript">getQ(544, 1179, 2406, 1180, 2407, 1181, 543, 2408, 9025, 9026);</script>
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</div>
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<div class="related"><a href="function.html">What is a Function?</a><a href="../algebra/index.html">Algebra Index</a></div>
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