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      <text>11.5.1 The meaning of a function of a function</text>
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      <text><![CDATA[<h4>The meaning of a function of a function</h4>

When we use a function like \(\sin 2x\) or \(e^{\ln x}\) or \(\sqrt{x^2 + 1}\) we are in fact dealing with a composite function or <b>function of a function</b>.
<p></p>
\(\sin 2x\) is the sine function of \(2x\). This is, in fact, how we `read' it:
<p></p>
\( \sin 2x \) is read `sine of \(2x\)'.
<p></p>
Similarly \(e^{\ln x}\) is the exponential function of the logarithm of \(x\):
<p></p>
\( e^{\ln x}\) is read `\(e\) to the power of \(\ln x\)'.
<p></p>
Finally \(\sqrt{x^2 + 1}\) is also a composite function. It is the square root function of the polynomial \(x^2 +1\):
<p></p>
\(\sqrt{x^2 + 1}\) is read as the `square root of \( (x^2 + 1) \)'.
<p></p>
When we talk about a function of a function in a general setting we will use the notation \(f(g(x))\) where both \(f\) and \(g\) are functions.

<hr>
<h4 class="HELM_example">Example</h4>
<p>Specify the functions \(f\), \(g\) for the composite functions \(f(g(x))\).</p>
<ol class="HELM_parts_inline">
    <li>\(\sin 2x\) </li>
    <li>\( e^{\ln x}\)</li>
    <li>\(\sqrt{x^2 + 1}\) </li>
</ol>

<h4 class="HELM_solution">Solution</h4>
<ol class="HELM_parts">
    <li>Here \(f\) is the sine function and \(g\) is the polynomial \(2x\). We often write
        <p></p>
        \(f(g) = \sin g\) and \( g(x) = 2x\) </li>
    <li>Here \(f(g) = e^g\) and \(g(x) = \ln x\)</li>
    <li>Here \(f(g) = \sqrt{g}\) and \(g(x) = x^2 + 1\)</li>
</ol>
<p>&nbsp;</p>
<hr>
<p>In each case the original function of \(x\) is obtained when \(g(x)\) is substituted into \(f(g)\).</p>]]></text>
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