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398 lines
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<h1 align="center">Inverse Functions</h1>
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<p align="center"><i>An inverse function goes the other way!</i></p>
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<p>Let us start with an example:</p>
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<p class="center">Here we have the function <b>f(x) = 2x+3</b>, written as a flow diagram:</p>
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<p class="center"><img src="images/flow-2xp3.svg" alt="2x+3" /></p>
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<p class="center">The <b>Inverse Function</b> goes the other way:</p>
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<p class="center"><img src="images/flow-2xp3-inv.svg" alt="Inverse" /></p>
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<p class="center">So the inverse of: 2x+3 is: (y-3)/2</p>
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<p> </p>
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<p>The inverse is usually shown by putting a little "-1" after the function name, like this: </p>
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<p align="center"> <span class="largest">f<sup>-1</sup>(y)</span></p>
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<p align="center"><i>We say "<b>f inverse</b> of y"</i></p>
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<p>So, the inverse of <span class="large">f(x) = 2x+3</span> is written:</p>
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<p align="center"> <span class="large">f<sup>-1</sup>(y) = (y-3)/2</span></p>
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<p>(I also used <b>y</b> instead of <b>x</b> to show that we are using a different value.) </p>
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<h2>Back to Where We Started</h2>
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<p>The cool thing about the inverse is that it should give us back the original value:</p>
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<p align="center"><img src="images/function-apple-banana.svg" alt="banana f() to apple f()-inverse back to banana" />
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<br /> When the function <span class="large">f</span> turns the apple into a banana,
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<br /> Then the <b>inverse</b> function <span class="large">f<sup>-1</sup></span> turns the banana back to the apple</p>
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<br />
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<div class="example">
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<h3>Example: </h3>
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<p>Using the formulas from above, we can start with <span class="large">x=4</span>:</p>
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<p align="center"> <span class="large">f(4) = 2×4+3 = 11</span></p>
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<p>We can then use the inverse on the 11:</p>
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<p align="center"><span class="large">f<sup>-1</sup>(11) = (11-3)/2 = 4</span></p>
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<p>And we magically get <b>4</b> back again!</p>
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<p>We can write that in one line:</p>
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<p align="center"><span class="large">f<sup>-1</sup>( f(4) ) = 4</span></p>
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<p class="center"><i>"f inverse of f of 4 equals 4"</i></p>
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</div>
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<p>So applying a function <span class="large">f</span> and then its inverse <span class="large">f<sup>-1</sup></span> gives us the original value back again:</p>
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<p align="center"><span class="large">f<sup>-1</sup>( f(x) ) = x</span></p>
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<p>We could also have put the functions in the other order and it still works:</p>
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<p align="center"><span class="large">f( f<sup>-1</sup>(x) ) = x</span></p>
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<div class="example">
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<h3>Example: </h3>
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<p>Start with:</p>
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<p align="center"><span class="large">f<sup>-1</sup>(11) = (11-3)/2 = 4</span></p>
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<p>And then:</p>
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<p align="center"> <span class="large">f(4) = 2×4+3 = 11</span></p>
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<p>So we can say:</p>
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<p align="center"><span class="large">f( f<sup>-1</sup>(11) ) = 11</span></p>
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<p class="center"><i>"f of f inverse of 11 equals 11"</i></p>
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</div>
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<h2>Solve Using Algebra</h2>
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<p>We can work out the inverse using Algebra. <b>Put "y" for "f(x)" and solve for x:</b></p>
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<table border="0" align="center">
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<tr>
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<td align="right">The function:</td>
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<td> </td>
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<td align="right"><b>f(x)</b></td>
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<td align="center"> = </td>
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<td align="left"><b>2x+3</b></td>
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</tr>
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<tr>
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<td align="right">Put "y" for "f(x)":</td>
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<td> </td>
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<td align="right"><b>y</b></td>
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<td align="center">=</td>
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<td align="left"><b>2x+3</b></td>
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</tr>
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<tr>
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<td align="right">Subtract 3 from both sides:</td>
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<td> </td>
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<td align="right"><b>y-3</b></td>
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<td align="center">=</td>
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<td align="left"><b>2x</b></td>
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</tr>
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<tr>
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<td align="right">Divide both sides by 2:</td>
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<td> </td>
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<td align="right"><b>(y-3)/2</b></td>
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<td align="center">=</td>
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<td align="left"><b>x</b></td>
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</tr>
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<tr>
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<td align="right">Swap sides:</td>
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<td> </td>
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<td align="right"><b>x</b></td>
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<td align="center">=</td>
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<td align="left"><b>(y-3)/2</b></td>
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</tr>
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<tr>
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<td align="right"> </td>
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<td> </td>
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<td align="right"> </td>
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<td align="center"> </td>
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<td align="left"> </td>
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</tr>
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<tr>
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<td align="right">Solution (put "f<sup>-1</sup>(y)" for "x") : </td>
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<td> </td>
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<td align="right"><b>f<sup>-1</sup>(y)</b></td>
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<td align="center">=</td>
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<td align="left"><b>(y-3)/2</b></td>
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</tr>
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</table>
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<p>This method works well for more difficult inverses.</p>
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<h2>Fahrenheit to Celsius</h2>
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<p>A useful example is converting between <a href="../temperature-conversion.html">Fahrenheit and Celsius</a>:</p>
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<div class="tbl">
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<div class="row"><span class="left">To convert Fahrenheit to Celsius:</span><span class="right"><b>f(F) = (F - 32) × <span class="intbl"><em>5</em><strong>9</strong></span></b></span></div>
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<div class="row"><span class="left">The <b>Inverse Function</b> (Celsius back to Fahrenheit):</span><span class="right"><b>f<sup>-1</sup>(C) = (C × <span class="intbl"><em>9</em><strong>5</strong></span>) + 32</b></span></div>
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</div>
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<p>For you: see if you can do the steps to create that inverse!</p>
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<h2>Inverses of Common Functions</h2>
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<p>It has been easy so far, because we know the inverse of Multiply is Divide, and the inverse of Add is Subtract, but what about other functions?</p>
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<p>Here is a list to help you:</p>
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<table border="0" align="center" cellspacing="10">
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<tr bgcolor="#FFFFCC">
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<td colspan="3" align="center" class="large">Inverses</td>
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<td width="220" align="center" class="large">Careful!</td>
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</tr>
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<tr>
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<td width="130" align="center" class="large"><img src="../numbers/images/40x40-add.gif" alt="add" width="40" height="40" border="0" /></td>
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<td width="60" align="center"><=></td>
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<td width="130" align="center" class="large"><img src="../numbers/images/40x40-sub.gif" alt="Subtract" width="40" height="40" border="0" /></td>
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<td width="220" align="center"> </td>
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</tr>
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<tr>
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<td width="130" align="center" class="large"><img src="../numbers/images/40x40-mult.gif" alt="Multiply" width="40" height="40" border="0" /></td>
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<td width="60" align="center"><=></td>
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<td width="130" align="center" class="large"><img src="../numbers/images/40x40-div.gif" alt="Divide" width="40" height="40" border="0" /></td>
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<td width="220" align="center">Don't divide by zero</td>
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</tr>
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<tr>
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<td width="130" align="center" class="large"><span class="intbl"><em>1</em><strong>x</strong></span></td>
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<td width="60" align="center"><=></td>
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<td width="130" align="center" class="large"><span class="intbl"><em>1</em><strong>y</strong></span></td>
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<td width="220" align="center">x and y not zero</td>
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</tr>
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<tr>
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<td width="130" align="center" class="large">x<sup>2</sup></td>
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<td width="60" align="center"><=></td>
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<td width="130" align="center" class="large"><img src="images/sqrt-y.gif" alt="sqrt-y" width="27" height="21" /></td>
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<td width="220" align="center">x and y ≥ 0</td>
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</tr>
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<tr>
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<td width="130" align="center" class="large">x<sup>n</sup></td>
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<td width="60" align="center"><=></td>
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<td width="130" align="center" class="large"><img src="images/n-root-y.gif" alt="nth-root-y" width="27" height="21" align="absmiddle" /> <b>or</b> <img src="images/y-1-n.gif" alt="y^(1/n)" width="20" height="26" align="absmiddle" /></td>
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<td width="220" align="center">n not zero
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<br /> <i>(different rules
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when n is odd, even, negative or positive)</i> </td>
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</tr>
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<tr>
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<td width="130" align="center" class="large">e<sup>x</sup></td>
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<td width="60" align="center"><=></td>
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<td width="130" align="center" class="large">ln(y)</td>
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<td width="220" align="center">y > 0</td>
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</tr>
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<tr>
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<td width="130" align="center" class="large">a<sup>x</sup></td>
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<td width="60" align="center"><=></td>
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<td width="130" align="center" class="large">log<sub>a</sub>(y)</td>
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<td width="220" align="center">y and a > 0</td>
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</tr>
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<tr>
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<td width="130" align="center" class="large">sin(x)</td>
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<td width="60" align="center"><=></td>
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<td width="130" align="center" class="large">sin<sup>-1</sup>(y)</td>
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<td width="220" align="center">-<span class="times">π</span>/2 to +<span class="times">π</span>/2</td>
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</tr>
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<tr>
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<td width="130" align="center" class="large">cos(x)</td>
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<td width="60" align="center"><=></td>
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<td width="130" align="center" class="large">cos<sup>-1</sup>(y)</td>
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<td width="220" align="center">0 to <span class="times">π</span></td>
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</tr>
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<tr>
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<td width="130" align="center" class="large">tan(x)</td>
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<td width="60" align="center"><=></td>
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<td width="130" align="center" class="large">tan<sup>-1</sup>(y)</td>
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<td width="220" align="center">-<span class="times">π</span>/2 to +<span class="times">π</span>/2</td>
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</tr>
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</table>
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<p>(Note: you can read more about <a href="../algebra/trig-inverse-sin-cos-tan.html">Inverse Sine, Cosine and Tangent</a>.)</p>
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<h2>Careful!</h2>
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<p>Did you see the "Careful!" column above? That is because some inverses work <b>only with certain values</b>.</p>
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<div class="example">
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<h3>Example: Square and Square Root</h3>
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<p>When we square a <b>negative</b> number, and then do the inverse, this happens:</p>
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<div class="tbl">
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<div class="row"><span class="left">Square:</span><span class="right">(<span class="hilite">−2</span>)<sup>2</sup> = 4</span></div>
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<div class="row"><span class="left">Inverse (Square Root): </span><span class="right">√(4) = <span class="hilite">2</span></span></div>
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</div>
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<p>But we didn't get the original value back! We got <b>2</b> instead of <b>−2</b>. Our fault for not being careful!</p>
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<p>So the square function (as it stands) <b>does not have an inverse</b></p>
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</div>
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<h3>But we can fix that!</h3>
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<p><b>Restrict the <a href="domain-range-codomain.html">Domain</a></b> (the values that can go into a function).</p>
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<div class="example">
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<h3>Example: (continued)</h3>
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<p>Just make sure we don't use negative numbers.</p>
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<p>In other words, restrict it to <b>x ≥ 0</b> and then we can have an inverse.</p>
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<p>So we have this situation:</p>
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<ul>
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<li><span class="large">x<sup>2</sup></span> does<b> not</b> have an inverse</li>
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<li>but <span class="large">{x<sup>2</sup> | x ≥ 0 }</span> (which says "x squared such that x is greater than or equal to zero" using <a href="set-builder-notation.html">set-builder notation</a>) <b>does </b>have an inverse.</li>
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</ul>
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</div>
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<h2>No Inverse?</h2>
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<p>Let us see graphically what is going on here:</p>
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<p align="center" class="larger">To be able to have an inverse we need <b>unique values</b>. </p>
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<p>Just think ... if there are two or more <b>x-values</b> for one <b>y-value</b>, how do we know which one to choose when going back?</p>
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<table border="0" align="center">
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<tr align="center">
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<td align="center"><span class="large">General Function</span></td>
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</tr>
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<tr>
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<td width="230" align="center"><img src="images/function-general-graph.svg" alt="General Function" /></td>
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</tr>
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<tr align="center">
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<td align="center" class="larger"><img src="../images/style/no.svg" alt="not" height="46" style="vertical-align:middle;" />No Inverse</td>
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|
</tr>
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|
</table>
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|
<p>Imagine we came <b>from</b> x<sub>1</sub> to a particular y value, where do we go back to? x<sub>1</sub> or x<sub>2</sub>? </p>
|
|
<p>In that case we can't have an inverse.</p>
|
|
<p>But if we can have exactly one x for every y we can have an inverse. </p>
|
|
<p>It is called a "one-to-one correspondence" or <a href="injective-surjective-bijective.html">Bijective</a>, like this</p>
|
|
<table border="0" align="center">
|
|
<tr align="center">
|
|
<td align="center" class="large">Bijective Function</td>
|
|
</tr>
|
|
<tr>
|
|
<td width="230" align="center"><img src="images/function-bijective-graph.svg" alt="Bijective Function" /></td>
|
|
</tr>
|
|
<tr align="center">
|
|
<td align="center" class="larger"><img src="../images/style/yes.svg" alt="not" height="46" style="vertical-align:middle;" />Has an Inverse</td>
|
|
</tr>
|
|
</table>
|
|
|
|
|
|
|
|
<div class="def">
|
|
<p class="larger">A function has to be "Bijective" to have an inverse.</p>
|
|
</div>
|
|
<p>So a bijective function follows stricter rules than a general function, which allows us to have an inverse.</p>
|
|
<h2>Domain and Range</h2>
|
|
<p>So what is all this talk about "<b>Restricting the Domain</b>"? </p>
|
|
<p style="float:left; margin: 0 10px 5px 0;"><img src="images/range-domain-graph.svg" alt="domain and range graph" /></p>
|
|
<p>In its simplest form the <b>domain</b> is all the values that go into a function (and the <b>range</b> is all the values that come out).</p>
|
|
<div style="clear:both"></div> <p>As it stands the function above does <b>not</b> have an inverse, because some y-values will have more than one x-value.</p>
|
|
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/range-domain-injective.svg" alt="doman and range" /></p>
|
|
<p>But we could restrict the domain so there is a <b>unique x for every y</b> ... </p><div style="clear:both"></div>
|
|
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/range-domain-inverse.svg" alt="doman and range" /></p>
|
|
<div align="right" class="larger">... and now we <b>can</b> have an inverse: </div>
|
|
<p>Note also:</p>
|
|
<ul>
|
|
<li>The function <b>f(x)</b> goes from the domain to the range,</li>
|
|
<li>The inverse function <b>f<sup>-1</sup>(y)</b> goes from the range back to the domain.</li>
|
|
</ul>
|
|
<p> </p>
|
|
<p> </p>
|
|
<p>Let's plot them both in terms of <b>x</b> ... so it is now <b>f<sup>-1</sup>(x)</b>, not <b>f<sup>-1</sup>(y)</b>:</p>
|
|
<p class="center"><img src="images/range-domain-flip.gif" alt="doman and range" width="223" height="193" /></p>
|
|
<p align="center" class="larger"><b>f(x)</b> and <b>f<sup>-1</sup>(x)</b> are like mirror images
|
|
<br /> (flipped about the diagonal). </p>
|
|
<p>In other words:</p>
|
|
<div class="def">
|
|
<p>The graph of <b>f(x)</b> and<b> f<sup>-1</sup>(x)</b> are symmetric across the line <b>y=x</b></p>
|
|
</div>
|
|
<p> </p>
|
|
<div class="example">
|
|
<h3>Example: Square and Square Root (continued)</h3>
|
|
<p><b>First</b>, we restrict the Domain to <b>x ≥ 0</b>:</p>
|
|
<ul>
|
|
<li><b>{x<sup>2</sup> | x ≥ 0 }</b> <i>"x squared such that x is greater than or equal to zero"</i></li>
|
|
<li><b>{√x | x ≥ 0 }</b> <i>"square root of x such that x is greater than or equal to zero"</i></li>
|
|
</ul>
|
|
<p> </p>
|
|
<p align="center"><img src="images/x2-vs-rootx.gif" alt="x^2 vs square root x" width="188" height="188" />
|
|
<br /> And you can see they are "mirror images"
|
|
<br /> of each other about the diagonal y=x.</p>
|
|
<p> </p>
|
|
<p style="float:right; margin: 0 0 5px 10px;"><img src="images/x2-vs-mrootx.gif" alt="x^2 vs -square root x" width="147" height="148" /></p>
|
|
<p>Note: when we restrict the domain to <b>x ≤ 0</b> (less than or equal to 0) the inverse is then <b>f<sup>-1</sup>(x) = −√x</b>:</p>
|
|
<ul>
|
|
<li><b>{x<sup>2</sup> | x ≤ 0 }</b></li>
|
|
<li><b>{−√x | x ≥ 0 }</b></li>
|
|
</ul>
|
|
<p>Which are inverses, too. </p>
|
|
</div>
|
|
<h2>Not Always Solvable!</h2>
|
|
<p>It is sometimes not possible to find an Inverse of a Function.</p>
|
|
<div class="example">
|
|
<p>Example: <span class="large">f(x) = x/2 + sin(x)</span></p>
|
|
<p>We cannot work out the inverse of this, because we cannot solve for "x":</p>
|
|
<p align="center"><span class="large">y = x/2 + sin(x)</span></p>
|
|
<p align="center"><span class="large">y ... ? = x</span></p>
|
|
</div>
|
|
<h2>Notes on Notation</h2>
|
|
<p>Even though we write <span class="large">f<sup>-1</sup>(x)</span>, the "-1" is <b>not </b>an <a href="../exponent.html">exponent (or power)</a>:</p>
|
|
<div class="beach">
|
|
<table border="0" align="center">
|
|
<tr>
|
|
<td width="150" align="center"><span class="large">f<sup>-1</sup>(x)</span></td>
|
|
<td width="150" align="center"><i><b>...is different to...</b></i></td>
|
|
<td width="150" align="center"><span class="large">f(x)</span><span class="large"><sup>-1</sup></span></td>
|
|
</tr>
|
|
<tr>
|
|
<td width="150" align="center">Inverse of the function <b>f</b></td>
|
|
<td width="150" align="center"> </td>
|
|
<td width="150" align="center"><b>f(x)<sup>-1</sup> = 1/f(x)</b>
|
|
<br /> (the <a href="../algebra/reciprocal.html">Reciprocal</a>) </td>
|
|
</tr>
|
|
</table>
|
|
</div>
|
|
<h2>Summary</h2>
|
|
<ul>
|
|
<div class="bigul">
|
|
<li>The inverse of f(x) is f<sup>-1</sup>(y)</li>
|
|
<li>We can find an inverse by reversing the "flow diagram"</li>
|
|
<li>Or we can find an inverse by using Algebra:
|
|
<ul>
|
|
<li>Put "y" for "f(x)", and </li>
|
|
<li>Solve for x</li>
|
|
</ul>
|
|
</li>
|
|
<li>We may need to <b>restrict the domain</b> for the function to have an inverse</li>
|
|
</div>
|
|
</ul>
|
|
<div class="questions">
|
|
<script type="text/javascript">
|
|
getQ(565, 566, 1197, 1198, 2357, 8417, 1199, 1200, 2358, 8418);
|
|
</script> </div>
|
|
<div class="related"><a href="function.html">What is A Function?</a> <a href="injective-surjective-bijective.html">Injective, Surjective and Bijective</a> <a href="sets-introduction.html">Sets</a></div>
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