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+    "markdown": "# The Inverse of a Function\n\n\n\nIn this section we will use these add-on packages:\n\n``` {.julia .cell-code}\nusing CalculusWithJulia\nusing Plots\n```\n\n\n\n\n---\n\n\nA (univariate) mathematical function relates or associates values of $x$ to values $y$ using the notation $y=f(x)$. A key point is a given $x$ is associated with just one $y$ value, though a given $y$ value may be associated with several different $x$ values. (Graphically, this is the vertical line test.)\n\n\nWe may conceptualize such a relation in many ways: through an algebraic rule; through the graph of $f;$ through a description of what $f$ 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 $x$ and outputs a value $y$. If a rule is given defining the function, the computation of $y$ is straightforward. A different question is not so easy: for a given value $y$ what value - or *values* - of $x$ (if any) produce an output of $y$? That is, what $x$ value(s) satisfy $f(x)=y$?\n\n\n*If* for each $y$ in some set of values there is just one $x$ value, then this operation associates to each value $y$ a single value $x$, so it too is a function. When that is the case we call this an *inverse* function.\n\n\nWhy is this useful? When available, it can help us solve equations. If we can write our equation as $f(x) = y$, then we can \"solve\" for $x$ through $x = g(y)$, where $g$ is this inverse function.\n\n\nLet's explore when we can \"solve\" for an inverse function.\n\n\nConsider the graph of the function $f(x) = 2^x$:\n\n::: {.cell hold='true' execution_count=4}\n``` {.julia .cell-code}\nf(x) = 2^x\nplot(f, 0, 4, legend=false)\nplot!([2,2,0], [0,f(2),f(2)])\n```\n\n::: {.cell-output .cell-output-display execution_count=5}\n![](inversefunctions_files/figure-html/cell-5-output-1.svg){}\n:::\n:::\n\n\nThe graph of a function is a representation of points $(x,f(x))$, so to *find* $f(c)$ from the graph, we begin on the $x$ axis at $c$, move vertically to the graph (the point $(c, f(c))$), and then move horizontally to the $y$ axis, intersecting it at $f(c)$. The figure shows this for $c=2$, from which we can read that $f(c)$ is about $4$. This is how an $x$ is associated to a single $y$.\n\n\nIf we were to *reverse* the direction, starting at $f(c)$ on the $y$ axis and then moving horizontally to the graph, and then vertically to the $x$-axis we end up at a value $c$ with the correct $f(c)$. This operation will form a function **if** the initial movement horizontally is guaranteed to find *no more than one* value on the graph. That is, to have an inverse function, there can not be two $x$ values corresponding to a given $y$ 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.\n\n\nMore formally, a function is called *one-to-one* *if* for any two $a \\neq b$, it must be that $f(a) \\neq f(b)$. Many functions are one-to-one, many are not. Familiar one-to-one functions are linear functions ($f(x)=a \\cdot x + b$ with $a\\neq 0$), odd powers of $x$ ($f(x)=x^{2k+1}$), and functions of the form $f(x)=x^{1/n}$ for $x \\geq 0$. In contrast, all *even* functions are *not* one-to-one, as $f(x) = f(-x)$ for any nonzero $x$ in the domain of $f$.\n\n\nA class of functions that are guaranteed to be one-to-one are the *strictly* increasing functions (which satisfy $a < b$ implies $f(a) < f(b)$). Similarly, strictly decreasing functions are one-to-one. The term strictly *monotonic* is used to describe either strictly increasing or strictly decreasing.  By the above observations, strictly monotonic function will have inverse functions.\n\n\nThe function $2^x$, graphed above, is strictly increasing, so it will have an inverse function. That is we can solve for $x$ in an equation like $2^x = 9$ using the inverse function of $f(x) = 2^x$, provided we can identify the inverse function.\n\n\n## How to solve for an inverse function?\n\n\nIf we know an inverse function exists, how can we find it?\n\n\nIf our function is given by a graph, the process above describes how to find the inverse function.\n\n\nHowever, typically we have a rule describing our function. What is the process then?  A simple example helps illustrate. The *linear* function $f(x) = 9/5\\cdot x + 32$ is strictly increasing, hence has an inverse function. What should it be? Let's describe the action of $f$: it multiplies $x$ by $9/5$ and then adds $32$. To \"invert\" this we *first* invert the adding of $32$ by subtracting $32$, then we would \"invert\" multiplying by $9/5$ by *dividing* by $9/5$. Hence $g(x)=(x-32)/(9/5)$. 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 **reverse** order. The reverse order might seem confusing, but this is hoLine truncated
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