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      <text>2.3.26 (Inverse Functions)</text>
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      <text><![CDATA[<h1>Inverse Functions</h1>

<p>Suppose we can remember that to convert a temperature given in Celsius to Farenheit, we multiply by nine, divide by five, and then add thirty-two.&nbsp; But what is the rule for converting back, from Farenheit to Celsius? How do we reverse this process?&nbsp; Is there a systematic method of doing this for other formulae?&nbsp; This is the question of finding the <b>inverse</b> to a function.</p>

<p>We often want to be able to "undo" a function, i.e. take a value \(y\) that we know is the output of a function \(f\), and find the value \(x\) which satisfies \(y=f(x)\).&nbsp; This is about finding the inverse of the function \(f\).&nbsp; We say that a function \(g\) is the <b>inverse </b>of \(f\) if \(g(f(x)) = x\)  for all \(x\) in the domain of \(f\).&nbsp; It follows that also \(f(g(y)) = y\) for all \(y\) in the range of \(f\).&nbsp; If a function \(f\) has an inverse, then we say that \(f\) is <b>invertible </b>and usually write this inverse as \(f^{-1}\). Note that we often use the notation \((g \circ f)(x) = g(f(x))\), which means apply the function \(f\) then apply the function \(g\). This is called the <b>composition</b> of the functions \(f\) and \(g\). Hence we say \(g\) is the inverse of \(f\) if \((g \circ f)(x) = id(x) = x\), where \(id\) is the identity function.</p>
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<p style="text-align: center;"><img src="@@PLUGINFILE@@/5_2_1_inverses.svg" alt="x and f(x) related by f and f inverse" width="254" height="143" class="img-responsive atto_image_button_text-bottom"></p>
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<p>Not every function has an inverse!&nbsp; You will already be familiar with this problem.&nbsp; The best-known example is the function \(f(x) = x^2\).&nbsp; What's the problem with this function?&nbsp; Two things: <br></p>
<ul>
<li>if \(y\) is a negative number, then no number \(x\) can be found so that \(f(x) = x^2 = y\); and</li>
<li>if \(y\) is a (strictly) positive number, then there is a choice of two values for \(x\) such that \(f(x) = x^2 = y\),  \(\sqrt{y}\) and \(-\sqrt{y}\).</li>
</ul>
<p>You will see that in some sense this is not a problem with the function itself, but more about the domain and range of the function. If we consider these more carefully, we might be able to find an inverse after all.</p>
<p>The range of the function \(y = f(x)\) is the set of values \(0 \leq y &lt; \infty\).&nbsp; So we have no hope of being able to define an inverse to \(f\) for other values of \(y\).&nbsp; Let's just accept this as a restriction we have to deal with: an inverse of a function is only defined on the range of that function.</p>
<p>We also have a problem with the domain.&nbsp; Whereas the range was too small, in some sense, the domain is too big: it gives us multiple options for our inverse.&nbsp; So we just consider the function \(f\) as being defined on a smaller domain, for example \(0 \leq x &lt; \infty\).&nbsp; Then for each value of \(y\) there is one <i>and only one</i> value of \(x\) in this smaller domain such that \(f(x) = y\).&nbsp; We have precisely excluded the negative values of \(x\) that would work.<br></p>
<p>This is the general situation: for a function to be invertible we need the following relationship between the function, domain, and range to be true: that for each point \(y\) in the range there is one and only one value of \(x\) such that \(f(x) = y\).&nbsp; As discussed above, this sometimes means "shrinking" the domain to a smaller set, and considering the function to be defined only on this smaller range of values.</p>
<p>In the case that this is true, and the inverse \(f^{-1}\) exists, then <br></p>
<ul>
<li>the domain of \(f^{-1}\) is the range of \(f\); and</li>
<li>the range of \(f^{-1}\) is the domain of \(f\).</li>
</ul>

<p>Suppose a function \(f\) is invertible.&nbsp; How do we find the inverse?&nbsp; We write \(y = f(x)\), and then try to rearrange to make \(x\) the subject of the equation.&nbsp; This then gives us a formula for the inverse of the function.</p>
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<h3>Example</h3>

<p>Given \(f(x) = 2x - 23\) find the inverse \(f^{-1}(x)\).</p>

<h3>Solution</h3>

<p>Set \(y = 2x - 23\) then adding \(23\) and dividing by two gives \(x = \frac{1}{2}(y +23)\). Therefore the inverse is \(f^{-1}(x) = \frac{1}{2}(x +23)\).</p>]]></text>
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