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<title>Evaluating Functions</title>
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<h1 align="center">Evaluating Functions</h1>
<h2>Evaluating Functions</h2>
<p>To evaluate a <a href="../sets/function.html">function</a> is to:</p>
<p align="center" class="larger">Replace (<a href="substitution.html">substitute</a>) its variable with a given number or expression.</p>
<p>Like in this example:</p>
<div class="example">
<h3>Example: evaluate the function <b>f(x) = 2x+4</b> for <b>x=5</b></h3>
<p>Just replace the variable &quot;x&quot; with &quot;5&quot;:</p>
<p align="center" class="larger">f(<span class="hi">5</span>) = 2&times;<span class="hi">5</span> + 4 = 14</p>
<p>Answer: <b>f(5) = 14</b></p>
</div>
<h2>More Examples</h2>
<p>Here is a function: </p>
<p align="center" class="larger">f(x) = 1 &minus; x + x<sup>2</sup></p>
<div class="center80">
<p>Important! The &quot;x&quot; is just a place-holder! And &quot;f&quot; is just a name.</p>
<p>These are all the <b>same function</b>:</p>
<ul>
<li>f(x) = 1 &minus; x + x<sup>2</sup></li>
<li>f(q) = 1 &minus; q + q<sup>2</sup></li>
<li>w(A) = 1 &minus; A + A<sup>2</sup></li>
<li>pumpkin(&theta;) = 1 &minus; &theta; + &theta;<sup>2</sup></li>
</ul>
</div>
<p>&nbsp;</p>
<h3>Evaluate For a Given Value:</h3>
<p>Let us evaluate that function for <span class="large">x=3</span>:</p>
<p align="center" class="larger"> f(<span class="hi">3</span>) = 1 &minus; <span class="hi">3</span> + <span class="hi">3</span><sup>2</sup> = 1 &minus; 3 + 9 = <b>7</b></p>
<h3>Evaluate For a Given Expression:</h3>
<p>Evaluating can also mean replacing with an <a href="definitions.html">expression</a> (such as <b>3m+1</b> or <b>v<sup>2</sup></b>).</p>
<p>Let us evaluate the function for<span class="large"> x=1/r</span>:</p>
<p align="center" class="larger"> f(<span class="hi">1/r</span>) = 1 &minus; (<span class="hi">1/r</span>) + (<span class="hi">1/r</span>)<sup>2</sup></p>
<p>Or evaluate the function for <span class="large">x = a&minus;4</span>: </p>
<div class="tbl">
<div class="row"><span class="left"><span class="larger">f(<span class="hi">a&minus;4</span>) </span>
</span><span class="right"><span class="larger">= 1 &minus; (<span class="hi">a&minus;4</span>) + (<span class="hi">a&minus;4</span>)<sup>2</sup></span>
</span></div>
<div class="row"><span class="left">&nbsp;</span><span class="right"><span class="larger">= 1 &minus; a + 4 + a<sup>2</sup> &minus; 8a + 16</span></span></div>
<div class="row"><span class="left">&nbsp;</span><span class="right"><span class="larger">= 21 &minus; 9a + a<sup>2</sup></span></span></div>
</div>
<h2>Another Example</h2>
<p>You can use your ability to evaluate functions to find other answers:</p>
<div class="example">
<h3>Example: h(x) = 3x<sup>2</sup> + ax &minus; 1</h3>
<ul>
<li>You are told that <b>h(3) = 8</b>, can you work out what &quot;a&quot; is?</li>
</ul>
<p>&nbsp;</p>
<div class="example2"></div>
<div class="tbl">
<div class="row"><span class="left">First, evaluate h(3):</span><span class="right">h(<span class="hi">3</span>) = 3&times;(<span class="hi">3</span>)<sup>2</sup> + a&times;<span class="hi">3</span> &minus; 1</span></div>
<div class="row"><span class="left">Simplify:</span><span class="right">h(3) = 27 + 3a &minus; 1</span></div>
<div class="row"><span class="left">&nbsp;</span><span class="right"><span class="larger">h(3) = 26 + 3a</span></span></div><p>&nbsp;</p>
<div class="row">
<span class="left"> Now ... we know that <b>h(3) = 8</b>, so:</span>
<span class="right"> 8 = 26 + 3a </span>
</div>
<div class="row"><span class="left">Swap sides:</span><span class="right">26 + 3a = 8</span></div>
<div class="row"><span class="left">Subtract 26 from both sides:</span><span class="right">3a = &minus;18</span></div>
<div class="row"><span class="left">Divide by 3:</span><span class="right">a = &minus;6</span></div>
</div>
<p>&nbsp;</p>
<p>Check:
h(3) = 3(3)<sup>2</sup> &minus; 6&times;3 &minus; 1 = 27 &minus; 18 &minus; 1 = 8 <img src="../images/style/yes.svg" alt="yes" /></p>
</div>
<h2>Careful!</h2>
<p>I recommend putting the substituted values inside parentheses <b>()</b> , so you don't make mistakes.</p>
<div class="example">
<h3>Example: evaluate the function <b>h(x) = x<sup>2</sup> + 2</b> for <b>x = &minus;3</b></h3>
<p>Replace the variable &quot;x&quot; with &quot;&minus;3&quot;:</p>
<p align="center">h(&minus;3) = <b>(&minus;3)</b><sup>2</sup> + 2 = <b>9 </b>+ 2 = <b>11</b></p>
<p>Without the () you could make a mistake:</p>
<p align="center">h(&minus;3) = <b>&minus;3<sup>2 </sup></b>+ 2 = <b>&minus;9</b> + 2 = <b>&minus;7</b> (WRONG!)</p>
</div>
<p>Also be careful of this:</p>
<p align="center" class="larger"><b>f(x+a)</b> is not the same as <b>f(x) + f(a)</b></p>
<div class="example">
<h3>Example: g(x) = x<sup>2</sup></h3>
<p align="center">g(w+1) = (w+1)<sup>2</sup> = <b>w<sup>2</sup> + 2w + 1</b></p>
<p align="center"><b><i>vs</i></b></p>
<p align="center">g(w) + g(1) = w<sup>2</sup> + 1<sup>2</sup> = <b>w<sup>2</sup> + 1</b></p>
<p>Different Result!</p>
</div>
<p>&nbsp;</p>
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