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175 lines
8.4 KiB
HTML
175 lines
8.4 KiB
HTML
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<title>Piecewise Functions</title>
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<h1 align="center">Piecewise Functions</h1>
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<h2>A Function Can be in Pieces</h2>
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<p>We can create functions that behave differently based on the input (x) value.</p>
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<p align="center"><img src="images/function-piecewise-a.gif" alt="Piecewise Function" width="243" height="192" /></p>
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<p align="center" class="larger">A function made up of 3 pieces</p>
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<p> </p>
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<div class="example">
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<h3>Example: </h3>
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<ul>
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<li>when x is less than 2, it gives <b>x<sup>2</sup></b>, </li>
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<li>when x is exactly 2 it gives <b>6</b></li>
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<li>when x is more than 2 and less than or equal to 6 it gives the line <b>10-x</b></li>
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</ul>
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<p>It looks like this:</p>
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<p align="center"><img src="images/function-piecewise-b.gif" alt="Piecewise Function" width="211" height="195" /></p>
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<p align="center">(a solid dot means "including",
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<br /> an open dot means "not including")</p>
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<p align="center"> </p>
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<p>And this is how we write it:</p>
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<p align="center"><img src="images/function-piecewise-b1.gif" alt="Piecewise Function" width="355" height="104" /></p>
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<p>The <a href="domain-range-codomain.html">Domain</a> (all the values that can go into the function) is all <a href="../numbers/real-numbers.html">Real Numbers</a> up to and including 6, which we can write like this:</p>
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<p align="center"><span class="large">Dom(f) = (-∞, 6] </span>(using <a href="intervals.html">Interval Notation</a>)</p>
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<p align="center"><span class="large">Dom(f) = <span class="large">{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 ≤ 6}</span> </span>(using <a href="set-builder-notation.html">Set Builder Notation</a>) </p>
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<p>And here are some example values:</p>
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<div align="center">
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<table width="30%" border="1" align="center">
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<tr>
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<th align="center">X</th>
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<th align="center">Y</th>
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</tr>
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<tr>
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<td align="center">−4</td>
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<td align="center">16</td>
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</tr>
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<tr>
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<td align="center">−2</td>
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<td align="center">4</td>
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</tr>
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<tr>
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<td align="center">0</td>
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<td align="center">0</td>
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</tr>
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<tr>
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<td align="center">1</td>
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<td align="center">1</td>
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</tr>
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<tr>
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<td align="center">2</td>
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<td align="center">6</td>
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</tr>
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<tr>
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<td align="center">3</td>
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<td align="center">7</td>
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</tr>
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</table>
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</div>
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</div>
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<p> </p>
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<div class="example">
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<h3>Example: Here is another piecewise function: </h3>
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<p class="center"><img src="../calculus/images/continuous-jump-fn.gif" width="223" height="77" alt="continuous jump fn h(x) = 2 if x<=1, x if x>1" /></p>
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<table border="0" align="center">
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<tr>
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<td>which looks like:</td>
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<td> </td>
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<td><img src="../calculus/images/continuous-jump-graph.gif" width="178" height="159" alt="continuous jump graph h(x)" /></td>
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</tr>
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</table>
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<p><span class="larger">What is h(−1)?</span></p>
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<p class="so">x is ≤ 1, so we use h(x) = 2, so <b>h(−1) = 2</b></p>
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<p><span class="larger">What is h(1)?</span></p>
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<p class="so">x is ≤ 1, so we use h(x) = 2, so <b>h(1) = 2</b></p>
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<p><span class="larger">What is h(4)?</span></p>
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<p class="so">x is > 1, so we use h(x) = x, so <b>h(4) = 4</b> </p>
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</div>
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<p>Piecewise functions let us make functions that do anything we want!</p>
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<div class="example">
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<h3>Example: A Doctor's fee is based on the length of time.</h3>
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<ul>
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<li>Up to 6 minutes costs $50</li>
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<li>Over 6 and up to 15 minutes costs $80</li>
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<li>Over 15 minutes costs $80 plus $5 per minute above 15 minutes</li>
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</ul>
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<p>Which we can write like this:</p>
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<p class="center"><img src="images/function-piecewise-d.gif" width="499" height="108" alt="function piecewise f(t)=50 if t<=6, 80 if t>6 and t<=15, 80 + 5(t-15) if t>15" /></p>
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<p> </p>
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<p>You visit for 12 minutes, what is the fee? <span class="hide">$80</span></p>
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<p>You visit for 20 minutes, what is the fee? <span class="hide">$80+$5(20-15) = $105</span></p>
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</div>
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<h2>The Absolute Value Function</h2>
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<p>The <a href="function-absolute-value.html">Absolute Value Function</a> is a famous Piecewise Function.</p>
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<p>It has two pieces:</p>
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<ul>
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<li>below zero: <b>-x</b></li>
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<li>from 0 onwards: <b>x</b></li>
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</ul>
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<p align="center"><img src="images/function-absolute.svg" alt="Absolute Value function" /></p>
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<p align="center"><span class="larger">f(x) = |x|</span></p>
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<p align="center"><img src="images/function-absolute-b.gif" alt="Absolute Value function" width="285" height="72" /></p>
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<h2>The Floor Function</h2>
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<p>The <a href="function-floor-ceiling.html">Floor Function</a> is a very special piecewise function. It has an infinite number of pieces:</p>
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<p align="center"><img src="images/function-floor-graph.svg" alt="Floor function" /></p>
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<p align="center" class="larger">The Floor Function</p>
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<p> </p>
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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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