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<li><a href="#how-to-solve-for-an-inverse-function" id="toc-how-to-solve-for-an-inverse-function" class="nav-link active" data-scroll-target="#how-to-solve-for-an-inverse-function"> <span class="header-section-number">10.1</span> How to solve for an inverse function?</a></li>
<li><a href="#functions-which-are-not-always-invertible" id="toc-functions-which-are-not-always-invertible" class="nav-link" data-scroll-target="#functions-which-are-not-always-invertible"> <span class="header-section-number">10.2</span> Functions which are not always invertible</a></li>
<li><a href="#formal-properties-of-the-inverse-function" id="toc-formal-properties-of-the-inverse-function" class="nav-link" data-scroll-target="#formal-properties-of-the-inverse-function"> <span class="header-section-number">10.3</span> Formal properties of the inverse function</a></li>
<li><a href="#the-graph-of-the-inverse-function" id="toc-the-graph-of-the-inverse-function" class="nav-link" data-scroll-target="#the-graph-of-the-inverse-function"> <span class="header-section-number">10.4</span> The graph of the inverse function</a>
<ul class="collapse">
<li><a href="#lines" id="toc-lines" class="nav-link" data-scroll-target="#lines"> <span class="header-section-number">10.4.1</span> Lines</a></li>
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<li><a href="#questions" id="toc-questions" class="nav-link" data-scroll-target="#questions"> <span class="header-section-number">10.5</span> Questions</a></li>
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<h1 class="title d-none d-lg-block"><span class="chapter-number">10</span>&nbsp; <span class="chapter-title">The Inverse of a Function</span></h1>
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<p>In this section we will use these add-on packages:</p>
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">CalculusWithJulia</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a><span class="im">using</span> <span class="bu">Plots</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<hr>
<p>A (univariate) mathematical function relates or associates values of <span class="math inline">\(x\)</span> to values <span class="math inline">\(y\)</span> using the notation <span class="math inline">\(y=f(x)\)</span>. A key point is a given <span class="math inline">\(x\)</span> is associated with just one <span class="math inline">\(y\)</span> value, though a given <span class="math inline">\(y\)</span> value may be associated with several different <span class="math inline">\(x\)</span> values. (Graphically, this is the vertical line test.)</p>
<p>We may conceptualize such a relation in many ways: through an algebraic rule; through the graph of <span class="math inline">\(f;\)</span> through a description of what <span class="math inline">\(f\)</span> does; or through a table of paired values, say. For the moment, let’s consider a function as rule that takes in a value of <span class="math inline">\(x\)</span> and outputs a value <span class="math inline">\(y\)</span>. If a rule is given defining the function, the computation of <span class="math inline">\(y\)</span> is straightforward. A different question is not so easy: for a given value <span class="math inline">\(y\)</span> what value - or <em>values</em> - of <span class="math inline">\(x\)</span> (if any) produce an output of <span class="math inline">\(y\)</span>? That is, what <span class="math inline">\(x\)</span> value(s) satisfy <span class="math inline">\(f(x)=y\)</span>?</p>
<p><em>If</em> for each <span class="math inline">\(y\)</span> in some set of values there is just one <span class="math inline">\(x\)</span> value, then this operation associates to each value <span class="math inline">\(y\)</span> a single value <span class="math inline">\(x\)</span>, so it too is a function. When that is the case we call this an <em>inverse</em> function.</p>
<p>Why is this useful? When available, it can help us solve equations. If we can write our equation as <span class="math inline">\(f(x) = y\)</span>, then we can “solve” for <span class="math inline">\(x\)</span> through <span class="math inline">\(x = g(y)\)</span>, where <span class="math inline">\(g\)</span> is this inverse function.</p>
<p>Let’s explore when we can “solve” for an inverse function.</p>
<p>Consider the graph of the function <span class="math inline">\(f(x) = 2^x\)</span>:</p>
<div class="cell" data-hold="true" data-execution_count="4">
<div class="sourceCode cell-code" id="cb2"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb2-1"><a href="#cb2-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fl">2</span><span class="op">^</span>x</span>
<span id="cb2-2"><a href="#cb2-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(f, <span class="fl">0</span>, <span class="fl">4</span>, legend<span class="op">=</span><span class="cn">false</span>)</span>
<span id="cb2-3"><a href="#cb2-3" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>([<span class="fl">2</span>,<span class="fl">2</span>,<span class="fl">0</span>], [<span class="fl">0</span>,<span class="fu">f</span>(<span class="fl">2</span>),<span class="fu">f</span>(<span class="fl">2</span>)])</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="5">
<p><img src="inversefunctions_files/figure-html/cell-5-output-1.svg" class="img-fluid"></p>
</div>
</div>
<p>The graph of a function is a representation of points <span class="math inline">\((x,f(x))\)</span>, so to <em>find</em> <span class="math inline">\(f(c)\)</span> from the graph, we begin on the <span class="math inline">\(x\)</span> axis at <span class="math inline">\(c\)</span>, move vertically to the graph (the point <span class="math inline">\((c, f(c))\)</span>), and then move horizontally to the <span class="math inline">\(y\)</span> axis, intersecting it at <span class="math inline">\(f(c)\)</span>. The figure shows this for <span class="math inline">\(c=2\)</span>, from which we can read that <span class="math inline">\(f(c)\)</span> is about <span class="math inline">\(4\)</span>. This is how an <span class="math inline">\(x\)</span> is associated to a single <span class="math inline">\(y\)</span>.</p>
<p>If we were to <em>reverse</em> the direction, starting at <span class="math inline">\(f(c)\)</span> on the <span class="math inline">\(y\)</span> axis and then moving horizontally to the graph, and then vertically to the <span class="math inline">\(x\)</span>-axis we end up at a value <span class="math inline">\(c\)</span> with the correct <span class="math inline">\(f(c)\)</span>. This operation will form a function <strong>if</strong> the initial movement horizontally is guaranteed to find <em>no more than one</em> value on the graph. That is, to have an inverse function, there can not be two <span class="math inline">\(x\)</span> values corresponding to a given <span class="math inline">\(y\)</span> value. This observation is often visualized through the “horizontal line test” - the graph of a function with an inverse function can only intersect a horizontal line at most in one place.</p>
<p>More formally, a function is called <em>one-to-one</em> <em>if</em> for any two <span class="math inline">\(a \neq b\)</span>, it must be that <span class="math inline">\(f(a) \neq f(b)\)</span>. Many functions are one-to-one, many are not. Familiar one-to-one functions are linear functions (<span class="math inline">\(f(x)=a \cdot x + b\)</span> with <span class="math inline">\(a\neq 0\)</span>), odd powers of <span class="math inline">\(x\)</span> (<span class="math inline">\(f(x)=x^{2k+1}\)</span>), and functions of the form <span class="math inline">\(f(x)=x^{1/n}\)</span> for <span class="math inline">\(x \geq 0\)</span>. In contrast, all <em>even</em> functions are <em>not</em> one-to-one, as <span class="math inline">\(f(x) = f(-x)\)</span> for any nonzero <span class="math inline">\(x\)</span> in the domain of <span class="math inline">\(f\)</span>.</p>
<p>A class of functions that are guaranteed to be one-to-one are the <em>strictly</em> increasing functions (which satisfy <span class="math inline">\(a &lt; b\)</span> implies <span class="math inline">\(f(a) &lt; f(b)\)</span>). Similarly, strictly decreasing functions are one-to-one. The term strictly <em>monotonic</em> is used to describe either strictly increasing or strictly decreasing. By the above observations, strictly monotonic function will have inverse functions.</p>
<p>The function <span class="math inline">\(2^x\)</span>, graphed above, is strictly increasing, so it will have an inverse function. That is we can solve for <span class="math inline">\(x\)</span> in an equation like <span class="math inline">\(2^x = 9\)</span> using the inverse function of <span class="math inline">\(f(x) = 2^x\)</span>, provided we can identify the inverse function.</p>
<section id="how-to-solve-for-an-inverse-function" class="level2" data-number="10.1">
<h2 data-number="10.1" class="anchored" data-anchor-id="how-to-solve-for-an-inverse-function"><span class="header-section-number">10.1</span> How to solve for an inverse function?</h2>
<p>If we know an inverse function exists, how can we find it?</p>
<p>If our function is given by a graph, the process above describes how to find the inverse function.</p>
<p>However, typically we have a rule describing our function. What is the process then? A simple example helps illustrate. The <em>linear</em> function <span class="math inline">\(f(x) = 9/5\cdot x + 32\)</span> is strictly increasing, hence has an inverse function. What should it be? Let’s describe the action of <span class="math inline">\(f\)</span>: it multiplies <span class="math inline">\(x\)</span> by <span class="math inline">\(9/5\)</span> and then adds <span class="math inline">\(32\)</span>. To “invert” this we <em>first</em> invert the adding of <span class="math inline">\(32\)</span> by subtracting <span class="math inline">\(32\)</span>, then we would “invert” multiplying by <span class="math inline">\(9/5\)</span> by <em>dividing</em> by <span class="math inline">\(9/5\)</span>. Hence <span class="math inline">\(g(x)=(x-32)/(9/5)\)</span>. We would generally simplify this, but let’s not for now. If we view a function as a composition of many actions, then we find the inverse by composing the inverse of these actions in <strong>reverse</strong> order. The reverse order might seem confusing, but this is how we get dressed and undressed: to dress we put on socks and then shoes. To undress we take off the shoes and then take off the socks.</p>
<p>When we solve algebraically for <span class="math inline">\(x\)</span> in <span class="math inline">\(y=9/5 \cdot x + 32\)</span> we do the same thing as we do verbally: we subtract <span class="math inline">\(32\)</span> from each side, and then divide by <span class="math inline">\(9/5\)</span> to isolate <span class="math inline">\(x\)</span>:</p>
<p><span class="math display">\[
\begin{align}
y &amp;= 9/5 \cdot x + 32\\
y - 32 &amp;= 9/5 \cdot x\\
(y-32) / (9/5) &amp;= x.
\end{align}
\]</span></p>
<p>From this, we have the function <span class="math inline">\(g(y) = (y-32) / (9/5)\)</span> is the inverse function of <span class="math inline">\(f(x) = 9/5\cdot x + 32\)</span>.</p>
<p><em>Usually</em> univariate functions are written with <span class="math inline">\(x\)</span> as the dummy variable, so it is typical to write <span class="math inline">\(g(x) = (x-32) / (9/5)\)</span> as the inverse function.</p>
<p><em>Usually</em> we use the name <span class="math inline">\(f^{-1}\)</span> for the inverse function of <span class="math inline">\(f\)</span>, so this would be most often <a href="http://tinyurl.com/qypbueb">seen</a> as <span class="math inline">\(f^{-1}(x) = (x-32)/(9/5)\)</span> or after simplification <span class="math inline">\(f^{-1}(x) = (5/9) \cdot (x-32)\)</span>.</p>
<div class="callout-note callout callout-style-default callout-captioned">
<div class="callout-header d-flex align-content-center">
<div class="callout-icon-container">
<i class="callout-icon"></i>
</div>
<div class="callout-caption-container flex-fill">
Note
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<p>The use of a negative exponent on the function name is <em>easily</em> confused for the notation for a reciprocal when it is used on a mathematical <em>expression</em>. An example might be the notation <span class="math inline">\((1/x)^{-1}\)</span>. As this is an expression this would simplify to <span class="math inline">\(x\)</span> and not the inverse of the <em>function</em> <span class="math inline">\(f(x)=1/x\)</span> (which is <span class="math inline">\(f^{-1}(x) = 1/x\)</span>).</p>
</div>
</div>
<section id="example" class="level5">
<h5 class="anchored" data-anchor-id="example">Example</h5>
<p>Suppose a transformation of <span class="math inline">\(x\)</span> is given by <span class="math inline">\(y = f(x) = (ax + b)/(cx+d)\)</span>. This function is invertible for most choices of the parameters. Find the inverse and describe it’s domain.</p>
<p>From the expression <span class="math inline">\(y=f(x)\)</span> we <em>algebraically</em> solve for <span class="math inline">\(x\)</span>:</p>
<p><span class="math display">\[
\begin{align*}
y &amp;= \frac{ax +b}{cx+d}\\
y \cdot (cx + d) &amp;= ax + b\\
ycx - ax &amp;= b - yd\\
(cy-a) \cdot x &amp;= b - dy\\
x &amp;= -\frac{dy - b}{cy-a}.
\end{align*}
\]</span></p>
<p>We see that to solve for <span class="math inline">\(x\)</span> we need to divide by <span class="math inline">\(cy-a\)</span>, so this expression can not be zero. So, using <span class="math inline">\(x\)</span> as the dummy variable, we have</p>
<p><span class="math display">\[
f^{-1}(x) = -\frac{dx - b}{cx-a},\quad cx-a \neq 0.
\]</span></p>
</section>
<section id="example-1" class="level5">
<h5 class="anchored" data-anchor-id="example-1">Example</h5>
<p>The function <span class="math inline">\(f(x) = (x-1)^5 + 2\)</span> is strictly increasing and so will have an inverse function. Find it.</p>
<p>Again, we solve algebraically starting with <span class="math inline">\(y=(x-1)^5 + 2\)</span> and solving for <span class="math inline">\(x\)</span>:</p>
<p><span class="math display">\[
\begin{align*}
y &amp;= (x-1)^5 + 2\\
y - 2 &amp;= (x-1)^5\\
(y-2)^{1/5} &amp;= x - 1\\
(y-2)^{1/5} + 1 &amp;= x.
\end{align*}
\]</span></p>
<p>We see that <span class="math inline">\(f^{-1}(x) = 1 + (x - 2)^{1/5}\)</span>. The fact that the power <span class="math inline">\(5\)</span> is an odd power is important, as this ensures a unique (real) solution to the fifth root of a value, in the above <span class="math inline">\(y-2\)</span>.</p>
</section>
<section id="example-2" class="level5">
<h5 class="anchored" data-anchor-id="example-2">Example</h5>
<p>The function <span class="math inline">\(f(x) = x^x, x \geq 1/e\)</span> is strictly increasing. However, trying to algebraically solve for an inverse function will quickly run into problems (without using specially defined functions). The existence of an inverse does not imply there will always be luck in trying to find a mathematical rule defining the inverse.</p>
</section>
</section>
<section id="functions-which-are-not-always-invertible" class="level2" data-number="10.2">
<h2 data-number="10.2" class="anchored" data-anchor-id="functions-which-are-not-always-invertible"><span class="header-section-number">10.2</span> Functions which are not always invertible</h2>
<p>Consider the function <span class="math inline">\(f(x) = x^2\)</span>. The graph - a parabola - is clearly not <em>monotonic</em>. Hence no inverse function exists. Yet, we can solve equations <span class="math inline">\(y=x^2\)</span> quite easily: <span class="math inline">\(y=\sqrt{x}\)</span> <em>or</em> <span class="math inline">\(y=-\sqrt{x}\)</span>. We know the square root undoes the squaring, but we need to be a little more careful to say the square root is the inverse of the squaring function.</p>
<p>The issue is there are generally <em>two</em> possible answers. To avoid this, we might choose to only take the <em>non-negative</em> answer. To make this all work as above, we restrict the domain of <span class="math inline">\(f(x)\)</span> and now consider the related function <span class="math inline">\(f(x)=x^2, x \geq 0\)</span>. This is now a monotonic function, so will have an inverse function. This is clearly <span class="math inline">\(f^{-1}(x) = \sqrt{x}\)</span>. (The <span class="math inline">\(\sqrt{x}\)</span> being defined as the principle square root or the unique <em>non-negative</em> answer to <span class="math inline">\(u^2-x=0\)</span>.)</p>
<p>The <a href="https://en.wikipedia.org/wiki/Inverse_function_theorem">inverse function theorem</a> basically says that if <span class="math inline">\(f\)</span> is <em>locally</em> monotonic, then an inverse function will exist <em>locally</em>. By “local” we mean in a neighborhood of <span class="math inline">\(c\)</span>.</p>
<section id="example-3" class="level5">
<h5 class="anchored" data-anchor-id="example-3">Example</h5>
<p>Consider the function <span class="math inline">\(f(x) = (1+x^2)^{-1}\)</span>. This bell-shaped function is even (symmetric about <span class="math inline">\(0\)</span>), so can not possibly be one-to-one. However, if the domain is restricted to <span class="math inline">\([0,\infty)\)</span> it is. The restricted function is strictly decreasing and its inverse is found, as follows:</p>
<p><span class="math display">\[
\begin{align*}
y &amp;= \frac{1}{1 + x^2}\\
1+x^2 &amp;= \frac{1}{y}\\
x^2 &amp;= \frac{1}{y} - 1\\
x &amp;= \sqrt{(1-y)/y}, \quad 0 \leq y \leq 1.
\end{align*}
\]</span></p>
<p>Then <span class="math inline">\(f^{-1}(x) = \sqrt{(1-x)/x}\)</span> where <span class="math inline">\(0 &lt; x \leq 1\)</span>. The somewhat complicated restriction for the the domain coincides with the range of <span class="math inline">\(f(x)\)</span>. We shall see next that this is no coincidence.</p>
</section>
</section>
<section id="formal-properties-of-the-inverse-function" class="level2" data-number="10.3">
<h2 data-number="10.3" class="anchored" data-anchor-id="formal-properties-of-the-inverse-function"><span class="header-section-number">10.3</span> Formal properties of the inverse function</h2>
<p>Consider again the graph of a monotonic function, in this case <span class="math inline">\(f(x) = x^2 + 2, x \geq 0\)</span>:</p>
<div class="cell" data-hold="true" data-execution_count="5">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> x<span class="op">^</span><span class="fl">2</span> <span class="op">+</span> <span class="fl">2</span></span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(f, <span class="fl">0</span>, <span class="fl">4</span>, legend<span class="op">=</span><span class="cn">false</span>)</span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>([<span class="fl">2</span>,<span class="fl">2</span>,<span class="fl">0</span>], [<span class="fl">0</span>,<span class="fu">f</span>(<span class="fl">2</span>),<span class="fu">f</span>(<span class="fl">2</span>)])</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="6">
<p><img src="inversefunctions_files/figure-html/cell-6-output-1.svg" class="img-fluid"></p>
</div>
</div>
<p>The graph is shown over the interval <span class="math inline">\((0,4)\)</span>, but the <em>domain</em> of <span class="math inline">\(f(x)\)</span> is all <span class="math inline">\(x \geq 0\)</span>. The <em>range</em> of <span class="math inline">\(f(x)\)</span> is clearly <span class="math inline">\(2 \leq x \leq \infty\)</span>.</p>
<p>The lines layered on the plot show how to associate an <span class="math inline">\(x\)</span> value to a <span class="math inline">\(y\)</span> value or vice versa (as <span class="math inline">\(f(x)\)</span> is one-to-one). The domain then of the inverse function is all the <span class="math inline">\(y\)</span> values for which a corresponding <span class="math inline">\(x\)</span> value exists: this is clearly all values bigger or equal to <span class="math inline">\(2\)</span>. The <em>range</em> of the inverse function can be seen to be all the images for the values of <span class="math inline">\(y\)</span>, which would be all <span class="math inline">\(x \geq 0\)</span>. This gives the relationship:</p>
<blockquote class="blockquote">
<p>the <em>range</em> of <span class="math inline">\(f(x)\)</span> is the <em>domain</em> of <span class="math inline">\(f^{-1}(x)\)</span>; furthermore the <em>domain</em> of <span class="math inline">\(f(x)\)</span> is the <em>range</em> for <span class="math inline">\(f^{-1}(x)\)</span>;</p>
</blockquote>
<p>From this we can see if we start at <span class="math inline">\(x\)</span>, apply <span class="math inline">\(f\)</span> we get <span class="math inline">\(y\)</span>, if we then apply <span class="math inline">\(f^{-1}\)</span> we will get back to <span class="math inline">\(x\)</span> so we have:</p>
<blockquote class="blockquote">
<p>For all <span class="math inline">\(x\)</span> in the domain of <span class="math inline">\(f\)</span>: <span class="math inline">\(f^{-1}(f(x)) = x\)</span>.</p>
</blockquote>
<p>Similarly, were we to start on the <span class="math inline">\(y\)</span> axis, we would see:</p>
<blockquote class="blockquote">
<p>For all <span class="math inline">\(x\)</span> in the domain of <span class="math inline">\(f^{-1}\)</span>: <span class="math inline">\(f(f^{-1}(x)) = x\)</span>.</p>
</blockquote>
<p>In short <span class="math inline">\(f^{-1} \circ f\)</span> and <span class="math inline">\(f \circ f^{-1}\)</span> are both identity functions, though on possibly different domains.</p>
</section>
<section id="the-graph-of-the-inverse-function" class="level2" data-number="10.4">
<h2 data-number="10.4" class="anchored" data-anchor-id="the-graph-of-the-inverse-function"><span class="header-section-number">10.4</span> The graph of the inverse function</h2>
<p>The graph of <span class="math inline">\(f(x)\)</span> is a representation of all values <span class="math inline">\((x,y)\)</span> where <span class="math inline">\(y=f(x)\)</span>. As the inverse flips around the role of <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span> we have:</p>
<blockquote class="blockquote">
<p>If <span class="math inline">\((x,y)\)</span> is a point on the graph of <span class="math inline">\(f(x)\)</span>, then <span class="math inline">\((y,x)\)</span> will be a point on the graph of <span class="math inline">\(f^{-1}(x)\)</span>.</p>
</blockquote>
<p>Let’s see this in action. Take the function <span class="math inline">\(2^x\)</span>. We can plot it by generating points to plot as follows:</p>
<div class="cell" data-hold="true" data-execution_count="6">
<div class="sourceCode cell-code" id="cb4"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb4-1"><a href="#cb4-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fl">2</span><span class="op">^</span>x</span>
<span id="cb4-2"><a href="#cb4-2" aria-hidden="true" tabindex="-1"></a>xs <span class="op">=</span> <span class="fu">range</span>(<span class="fl">0</span>, <span class="fl">2</span>, length<span class="op">=</span><span class="fl">50</span>)</span>
<span id="cb4-3"><a href="#cb4-3" aria-hidden="true" tabindex="-1"></a>ys <span class="op">=</span> <span class="fu">f</span>.(xs)</span>
<span id="cb4-4"><a href="#cb4-4" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(xs, ys, color<span class="op">=:</span>blue, label<span class="op">=</span><span class="st">"f"</span>)</span>
<span id="cb4-5"><a href="#cb4-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(ys, xs, color<span class="op">=:</span>red, label<span class="op">=</span><span class="st">"f⁻¹"</span>) <span class="co"># the inverse</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="7">
<p><img src="inversefunctions_files/figure-html/cell-7-output-1.svg" class="img-fluid"></p>
</div>
</div>
<p>By flipping around the <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span> values in the <code>plot!</code> command, we produce the graph of the inverse function - when viewed as a function of <span class="math inline">\(x\)</span>. We can see that the domain of the inverse function (in red) is clearly different from that of the function (in blue).</p>
<p>The inverse function graph can be viewed as a symmetry of the graph of the function. Flipping the graph for <span class="math inline">\(f(x)\)</span> around the line <span class="math inline">\(y=x\)</span> will produce the graph of the inverse function: Here we see for the graph of <span class="math inline">\(f(x) = x^{1/3}\)</span> and its inverse function:</p>
<div class="cell" data-hold="true" data-execution_count="7">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fu">cbrt</span>(x)</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a>xs <span class="op">=</span> <span class="fu">range</span>(<span class="op">-</span><span class="fl">2</span>, <span class="fl">2</span>, length<span class="op">=</span><span class="fl">150</span>)</span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a>ys <span class="op">=</span> <span class="fu">f</span>.(xs)</span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(xs, ys, color<span class="op">=:</span>blue, aspect_ratio<span class="op">=:</span>equal, legend<span class="op">=</span><span class="cn">false</span>)</span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(ys, xs, color<span class="op">=:</span>red)</span>
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(identity, color<span class="op">=:</span>green, linestyle<span class="op">=:</span>dash)</span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a>x, y <span class="op">=</span> <span class="fl">1</span><span class="op">/</span><span class="fl">2</span>, <span class="fu">f</span>(<span class="fl">1</span><span class="op">/</span><span class="fl">2</span>)</span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>([x,y], [y,x], color<span class="op">=:</span>green, linestyle<span class="op">=:</span>dot)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="8">
<p><img src="inversefunctions_files/figure-html/cell-8-output-1.svg" class="img-fluid"></p>
</div>
</div>
<p>We drew a line connecting <span class="math inline">\((1/2, f(1/2))\)</span> to <span class="math inline">\((f(1/2),1/2)\)</span>. We can see that it crosses the line <span class="math inline">\(y=x\)</span> perpendicularly, indicating that points are symmetric about this line. (The plotting argument <code>aspect_ratio=:equal</code> ensures that the <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span> axes are on the same scale, so that this type of line will look perpendicular.)</p>
<p>One consequence of this symmetry, is that if <span class="math inline">\(f\)</span> is strictly increasing, then so is its inverse.</p>
<p>!!!note In the above we used <code>cbrt(x)</code> and not <code>x^(1/3)</code>. The latter usage assumes that <span class="math inline">\(x \geq 0\)</span> as it isn’t guaranteed that for all real exponents the answer will be a real number. The <code>cbrt</code> function knows there will always be a real answer and provides it.</p>
<section id="lines" class="level3" data-number="10.4.1">
<h3 data-number="10.4.1" class="anchored" data-anchor-id="lines"><span class="header-section-number">10.4.1</span> Lines</h3>
<p>The slope of <span class="math inline">\(f(x) = 9/5 \cdot x + 32\)</span> is clearly <span class="math inline">\(9/5\)</span> and the slope of the inverse function <span class="math inline">\(f^{-1}(x) = 5/9 \cdot (x-32)\)</span> is clearly <span class="math inline">\(5/9\)</span> - or the reciprocal. This makes sense, as the slope is the rise over the run, and by flipping the <span class="math inline">\(x\)</span> and <span class="math inline">\(y\)</span> values we merely flip over the rise and the run.</p>
<p>Now consider the graph of the <em>tangent line</em> to a function. This concept will be better defined later, for now, it is a line “tangent” to the graph of <span class="math inline">\(f(x)\)</span> at a point <span class="math inline">\(x=c\)</span>.</p>
<p>For concreteness, we consider <span class="math inline">\(f(x) = \sqrt{x}\)</span> at <span class="math inline">\(c=2\)</span>. The tangent line will have slope <span class="math inline">\(1/(2\sqrt{2})\)</span> and will go through the point <span class="math inline">\((2, f(2)\)</span>. We graph the function, its tangent line, and their inverses:</p>
<div class="cell" data-hold="true" data-execution_count="8">
<div class="sourceCode cell-code" id="cb6"><pre class="sourceCode julia code-with-copy"><code class="sourceCode julia"><span id="cb6-1"><a href="#cb6-1" aria-hidden="true" tabindex="-1"></a><span class="fu">f</span>(x) <span class="op">=</span> <span class="fu">sqrt</span>(x)</span>
<span id="cb6-2"><a href="#cb6-2" aria-hidden="true" tabindex="-1"></a>c <span class="op">=</span> <span class="fl">2</span></span>
<span id="cb6-3"><a href="#cb6-3" aria-hidden="true" tabindex="-1"></a><span class="fu">tl</span>(x) <span class="op">=</span> <span class="fu">f</span>(c) <span class="op">+</span> <span class="fl">1</span><span class="op">/</span>(<span class="fl">2</span> <span class="op">*</span> <span class="fu">sqrt</span>(<span class="fl">2</span>)) <span class="op">*</span> (x <span class="op">-</span> c)</span>
<span id="cb6-4"><a href="#cb6-4" aria-hidden="true" tabindex="-1"></a>xs <span class="op">=</span> <span class="fu">range</span>(<span class="fl">0</span>, <span class="fl">3</span>, length<span class="op">=</span><span class="fl">150</span>)</span>
<span id="cb6-5"><a href="#cb6-5" aria-hidden="true" tabindex="-1"></a>ys <span class="op">=</span> <span class="fu">f</span>.(xs)</span>
<span id="cb6-6"><a href="#cb6-6" aria-hidden="true" tabindex="-1"></a>zs <span class="op">=</span> <span class="fu">tl</span>.(xs)</span>
<span id="cb6-7"><a href="#cb6-7" aria-hidden="true" tabindex="-1"></a><span class="fu">plot</span>(xs, ys, color<span class="op">=:</span>blue, legend<span class="op">=</span><span class="cn">false</span>)</span>
<span id="cb6-8"><a href="#cb6-8" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(xs, zs, color<span class="op">=:</span>blue) <span class="co"># the tangent line</span></span>
<span id="cb6-9"><a href="#cb6-9" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(ys, xs, color<span class="op">=:</span>red) <span class="co"># the inverse function</span></span>
<span id="cb6-10"><a href="#cb6-10" aria-hidden="true" tabindex="-1"></a><span class="fu">plot!</span>(zs, xs, color<span class="op">=:</span>red) <span class="co"># inverse of tangent line</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-display" data-execution_count="9">
<p><img src="inversefunctions_files/figure-html/cell-9-output-1.svg" class="img-fluid"></p>
</div>
</div>
<p>What do we see? In blue, we can see the familiar square root graph along with a “tangent” line through the point <span class="math inline">\((2, f(2))\)</span>. The red graph of <span class="math inline">\(f^{-1}(x) = x^2, x \geq 0\)</span> is seen and, perhaps surprisingly, a tangent line. This is at the point <span class="math inline">\((f(2), 2)\)</span>. We know the slope of this tangent line is the reciprocal of the slope of the red tangent line. This gives this informal observation:</p>
<blockquote class="blockquote">
<p>If the graph of <span class="math inline">\(f(x)\)</span> has a tangent line at <span class="math inline">\((c, f(c))\)</span> with slope <span class="math inline">\(m\)</span>, then the graph of <span class="math inline">\(f^{-1}(x)\)</span> will have a tangent line at <span class="math inline">\((f(c), c)\)</span> with slope <span class="math inline">\(1/m\)</span>.</p>
</blockquote>
<p>This is reminiscent of the formula for the slope of a perpendicular line, <span class="math inline">\(-1/m\)</span>, but quite different, as this formula implies the two lines have either both positive slopes or both negative slopes, unlike the relationship in slopes between a line and a perpendicular line.</p>
<p>The key here is that the shape of <span class="math inline">\(f(x)\)</span> near <span class="math inline">\(x=c\)</span> is somewhat related to the shape of <span class="math inline">\(f^{-1}(x)\)</span> at <span class="math inline">\(f(c)\)</span>. In this case, if we use the tangent line as a fill in for how steep a function is, we see from the relationship that if <span class="math inline">\(f(x)\)</span> is “steep” at <span class="math inline">\(x=c\)</span>, then <span class="math inline">\(f^{-1}(x)\)</span> will be “shallow” at <span class="math inline">\(x=f(c)\)</span>.</p>
</section>
</section>
<section id="questions" class="level2" data-number="10.5">
<h2 data-number="10.5" class="anchored" data-anchor-id="questions"><span class="header-section-number">10.5</span> Questions</h2>
<section id="question" class="level6">
<h6 class="anchored" data-anchor-id="question">Question</h6>
<p>Is it possible that a function have two different inverses?</p>
<div class="cell" data-hold="true" data-execution_count="9">
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<span class="label-body px-1">
No, for all \(x\) in the domain an an inverse, the value of any inverse will be the same, hence all inverse functions would be identical.
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Yes, the function \(f(x) = x^2, x \geq 0\) will have a different inverse than the same function \(f(x) = x^2, x \leq 0\)
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<section id="question-1" class="level6">
<h6 class="anchored" data-anchor-id="question-1">Question</h6>
<p>A function takes a value <span class="math inline">\(x\)</span> adds <span class="math inline">\(1\)</span>, divides by <span class="math inline">\(2\)</span>, and then subtracts <span class="math inline">\(1\)</span>. Is the function “one-to-one”?</p>
<div class="cell" data-hold="true" data-execution_count="10">
<div class="cell-output cell-output-display" data-execution_count="11">
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Yes, the function is the linear function \(f(x)=(x+1)/2 + 1\) and so is monotonic.
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</label>
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<input class="form-check-input" type="radio" name="radio_2876581345630770408" id="radio_2876581345630770408_2" value="2">
<span class="label-body px-1">
No, the function is \(1\) then \(2\) then \(1\), but not "one-to-one"
</span>
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</section>
<section id="question-2" class="level6">
<h6 class="anchored" data-anchor-id="question-2">Question</h6>
<p>Is the function <span class="math inline">\(f(x) = x^5 - x - 1\)</span> one-to-one?</p>
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No, a graph over \((-2,2)\) will show this.
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Yes, a graph over \((-100, 100)\) will show this.
</span>
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</div>
</div>
</section>
<section id="question-3" class="level6">
<h6 class="anchored" data-anchor-id="question-3">Question</h6>
<p>A function is given by the table</p>
<pre class="{verbatim}"><code>x | y
--------
1 | 3
2 | 4
3 | 5
4 | 3
5 | 4
6 | 5</code></pre>
<p>Is the function one-to-one?</p>
<div class="cell" data-hold="true" data-execution_count="12">
<div class="cell-output cell-output-display" data-execution_count="13">
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Yes
</span>
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<div class="form-check">
<label class="form-check-label" for="radio_18358563740150383766_2">
<input class="form-check-input" type="radio" name="radio_18358563740150383766" id="radio_18358563740150383766_2" value="2">
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No
</span>
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</section>
<section id="question-4" class="level6">
<h6 class="anchored" data-anchor-id="question-4">Question</h6>
<p>A function is defined by its graph.</p>
<div class="cell" data-hold="true" data-execution_count="13">
<div class="cell-output cell-output-display" data-execution_count="14">
<p><img src="inversefunctions_files/figure-html/cell-14-output-1.svg" class="img-fluid"></p>
</div>
</div>
<p>Over the domain shown, is the function one-to-one?</p>
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<input class="form-check-input" type="radio" name="radio_14928273129286464133" id="radio_14928273129286464133_1" value="1">
<span class="label-body px-1">
Yes
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_14928273129286464133_2">
<input class="form-check-input" type="radio" name="radio_14928273129286464133" id="radio_14928273129286464133_2" value="2">
<span class="label-body px-1">
No
</span>
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}
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</div>
</section>
<section id="question-5" class="level6">
<h6 class="anchored" data-anchor-id="question-5">Question</h6>
<p>Suppose <span class="math inline">\(f(x) = x^{-1}\)</span>.</p>
<p>What is <span class="math inline">\(g(x) = (f(x))^{-1}\)</span>?</p>
<div class="cell" data-hold="true" data-execution_count="15">
<div class="cell-output cell-output-display" data-execution_count="16">
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<input class="form-check-input" type="radio" name="radio_18046111943366457579" id="radio_18046111943366457579_1" value="1">
<span class="label-body px-1">
\(g(x) = x\)
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_18046111943366457579_2">
<input class="form-check-input" type="radio" name="radio_18046111943366457579" id="radio_18046111943366457579_2" value="2">
<span class="label-body px-1">
\(g(x) = x^{-1}\)
</span>
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</div>
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}
})});
</script>
</div>
</div>
<p>What is <span class="math inline">\(g(x) = f^{-1}(x)\)</span>?</p>
<div class="cell" data-hold="true" data-execution_count="16">
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<input class="form-check-input" type="radio" name="radio_6150813960233463858" id="radio_6150813960233463858_1" value="1">
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\(g(x) = x\)
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_6150813960233463858_2">
<input class="form-check-input" type="radio" name="radio_6150813960233463858" id="radio_6150813960233463858_2" value="2">
<span class="label-body px-1">
\(g(x) = x^{-1}\)
</span>
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</div>
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</section>
<section id="question-6" class="level6">
<h6 class="anchored" data-anchor-id="question-6">Question</h6>
<p>A function, <span class="math inline">\(f\)</span>, is given by its graph:</p>
<div class="cell" data-execution_count="17">
<div class="cell-output cell-output-display" data-execution_count="18">
<p><img src="inversefunctions_files/figure-html/cell-18-output-1.svg" class="img-fluid"></p>
</div>
</div>
<p>What is the value of <span class="math inline">\(f(1)\)</span>?</p>
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<p>What is the value of <span class="math inline">\(f^{-1}(1)\)</span>?</p>
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}
});
</script>
</div>
</div>
<p>What is the value of <span class="math inline">\((f(1))^{-1}\)</span>?</p>
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<input id="15187121442443939987" type="number" class="form-control" placeholder="Numeric answer">
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</div>
</div>
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if (explanation != null) {
explanation.style.display = "block";
}
}
});
</script>
</div>
</div>
<p>What is the value of <span class="math inline">\(f^{-1}(1/2)\)</span>?</p>
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<input id="312525117406379464" type="number" class="form-control" placeholder="Numeric answer">
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</div>
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</div>
</div>
</form>
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if (explanation != null) {
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}
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</div>
</div>
</section>
<section id="question-7" class="level6">
<h6 class="anchored" data-anchor-id="question-7">Question</h6>
<p>A function is described as follows: for <span class="math inline">\(x &gt; 0\)</span> it takes the square root, adds <span class="math inline">\(1\)</span> and divides by <span class="math inline">\(2\)</span>.</p>
<p>What is the inverse of this function?</p>
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<div class="cell-output cell-output-display" data-execution_count="23">
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<div class="form-check">
<label class="form-check-label" for="radio_17570845845840541456_1">
<input class="form-check-input" type="radio" name="radio_17570845845840541456" id="radio_17570845845840541456_1" value="1">
<span class="label-body px-1">
The function that takes square of the value, then subtracts \(1\), and finally multiplies by \(2\).
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_17570845845840541456_2">
<input class="form-check-input" type="radio" name="radio_17570845845840541456" id="radio_17570845845840541456_2" value="2">
<span class="label-body px-1">
The function that divides by \(2\), adds \(1\), and then takes the square root of the value.
</span>
</label>
</div>
<div class="form-check">
<label class="form-check-label" for="radio_17570845845840541456_3">
<input class="form-check-input" type="radio" name="radio_17570845845840541456" id="radio_17570845845840541456_3" value="3">
<span class="label-body px-1">
The function that multiplies by \(2\), subtracts \(1\) and then squares the value.
</span>
</label>
</div>
</div>
</div>
<div id="17570845845840541456_message" style="padding-bottom: 15px"></div>
</div>
</div>
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</section>
<section id="question-8" class="level6">
<h6 class="anchored" data-anchor-id="question-8">Question</h6>
<p>A function, <span class="math inline">\(f\)</span>, is specified by a table:</p>
<pre class="{verbatim}"><code>x | y
-------
1 | 2
2 | 3
3 | 5
4 | 8
5 | 13</code></pre>
<p>What is <span class="math inline">\(f(3)\)</span>?</p>
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<p>What is <span class="math inline">\(f^{-1}(3)\)</span>?</p>
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<p>What is <span class="math inline">\(f(5)^{-1}\)</span>?</p>
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<p>What is <span class="math inline">\(f^{-1}(5)\)</span>?</p>
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</section>
<section id="question-9" class="level6">
<h6 class="anchored" data-anchor-id="question-9">Question</h6>
<p>Find the inverse function of <span class="math inline">\(f(x) = (x^3 + 4)/5\)</span>.</p>
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\(f^{-1}(x) = (5y-4)^{1/3}\)
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\(f^{-1}(x) = 5/(x^3 + 4)\)
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\(f^{-1}(x) = (5y-4)^3\)
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</section>
<section id="question-10" class="level6">
<h6 class="anchored" data-anchor-id="question-10">Question</h6>
<p>Find the inverse function of <span class="math inline">\(f(x) = x^\pi + e, x \geq 0\)</span>.</p>
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\(f^{-1}(x) = (x-\pi)^{e}\)
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\(f^{-1}(x) = (x-e)^{\pi}\)
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\(f^{-1}(x) = (x-e)^{1/\pi}\)
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<section id="question-11" class="level6">
<h6 class="anchored" data-anchor-id="question-11">Question</h6>
<p>What is the <em>domain</em> of the inverse function for <span class="math inline">\(f(x) = x^2 + 7, x \geq 0\)</span>?</p>
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\([0, \infty)\)
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\((-\infty, \infty)\)
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\([7, \infty)\)
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<section id="question-12" class="level6">
<h6 class="anchored" data-anchor-id="question-12">Question</h6>
<p>What is the <em>range</em> of the inverse function for <span class="math inline">\(f(x) = x^2 + 7, x \geq 0\)</span>?</p>
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\([7, \infty)\)
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\((-\infty, \infty)\)
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\([0, \infty)\)
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</section>
<section id="question-13" class="level6">
<h6 class="anchored" data-anchor-id="question-13">Question</h6>
<p>From the plot, are blue and red inverse functions?</p>
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Yes
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<input class="form-check-input" type="radio" name="radio_14050108626791112652" id="radio_14050108626791112652_2" value="2">
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No
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<p>From the plot, are blue and red inverse functions?</p>
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<p><img src="inversefunctions_files/figure-html/cell-34-output-1.svg" class="img-fluid"></p>
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Yes
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<input class="form-check-input" type="radio" name="radio_11717651090467797062" id="radio_11717651090467797062_2" value="2">
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No
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<section id="question-14" class="level6">
<h6 class="anchored" data-anchor-id="question-14">Question</h6>
<p>The function <span class="math inline">\(f(x) = (ax + b)/(cx + d)\)</span> is known as a <a href="http://tinyurl.com/oemweyj">Mobius</a> transformation and can be expressed as a composition of <span class="math inline">\(4\)</span> functions, <span class="math inline">\(f_4 \circ f_3 \circ f_2 \circ f_1\)</span>:</p>
<ul>
<li>where <span class="math inline">\(f_1(x) = x + d/c\)</span> is a translation,</li>
<li>where <span class="math inline">\(f_2(x) = x^{-1}\)</span> is inversion and reflection,</li>
<li>where <span class="math inline">\(f_3(x) = ((bc-ad)/c^2) \cdot x\)</span> is scaling,</li>
<li>and <span class="math inline">\(f_4(x) = x + a/c\)</span> is a translation.</li>
</ul>
<p>For <span class="math inline">\(x=10\)</span>, what is <span class="math inline">\(f(10)\)</span>?</p>
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<p>For <span class="math inline">\(x=10\)</span>, what is <span class="math inline">\(f_4(f_3(f_2(f_1(10))))\)</span>?</p>
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<p>The last two answers should be the same, why?</p>
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As the latter is more complicated than the former.
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As \(f_4(f_3(f_2(f_1(x))))=(f_1 \circ f_2 \circ f_3 \circ f_4)(x)\)
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As \(f_4(f_3(f_2(f)_1(x))))=(f_4 \circ f_3 \circ f_2 \circ f_1)(x)\)
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<p>Let <span class="math inline">\(g_1\)</span>, <span class="math inline">\(g_2\)</span>, <span class="math inline">\(g_3\)</span>, and <span class="math inline">\(g_4\)</span> denote the inverse functions. Clearly, <span class="math inline">\(g_1(x) = x- d/c\)</span> and <span class="math inline">\(g+4(x) = x - a/c\)</span>, as the inverse of adding a constant is subtracting the constant.</p>
<p>What is <span class="math inline">\(g_2(x)=f_2^{-1}(x)\)</span>?</p>
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\(g_2(x) = x\)
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\(g_2(x) = x^{-1}\)
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\(g_2(x) = x -1\)
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<p>What is <span class="math inline">\(g_3(x)=f_3^{-1}(x)\)</span>?</p>
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\(c^2/(b\cdot c - a\cdot d) \cdot x\)
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\((b\cdot c-a\cdot d)/c^2 \cdot x\)
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\(c^2 x\)
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<p>Given these, what is the value of <span class="math inline">\(g_4(g_3(g_2(g_1(f_4(f_3(f_2(f_1(10))))))))\)</span>?</p>
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<p>What about the value of <span class="math inline">\(g_1(g_2(g_3(g_4(f_4(f_3(f_2(f_1(10))))))))\)</span>?</p>
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