Archive for the ‘Elementary Functions’ Category
April 29th, 2009 | 
Author: Graphing functions is straightforward you make a table of inputs and outputs for your function. You draw the points on graph paper at the different coordinates. A picture of the function will emerge if you plot enough points from the domain. When you are fairly confident that a picture emerges you can draw a smooth line between the points according to what you believe the intermediate values are, called an interpolation.
For any function the inverse can be found graphically. Consider the inverse function pair $e^x$ and $\ln x$. We know that $e^{\ln x} = x$ and $\ln(e^x)=x$ by the definition of inverses. By plotting the two functions we see we see that they are mirror images of each other about the line $y = x$. This is true for any function-inverse pair. Drawing the mirror image about $y=x $ will generate the inverse function for you.
\begin{figure}[htp]
\begin{center}
\includegraphics[width= 3in]{../images/expln.pdf}
\caption{A function inverse pair is symmetric about $y=x$}
\end{center}
\end{figure}
In order that what you have plotted is to be an actual function it must pass the vertical line test. If you draw a vertical line at any $x$ coordinate then it must only cross the function twice. If it crosses twice the function would have two outputs for the same input which is not allowed. An inverse function is a function which is drawn as the mirror image along $y=x$. For a function to have an inverse it must pass what is called the horizontal line test. Any horizontal line must only cross the function once, otherwise the mirror image would not pass the vertical line test. Remember the vertical line test is for testing whether a function is an actual function, and the horizontal line test is for testing whether a function has an inverse.
There are a number of transformations that can be applied to a function to get a new function that can be plotted without additional knowledge but some transformation rules. These transformations include a shift to the left or right, a shift up or down, horizontal or vertical stretching or compressions, and reflections about either axis. Understanding these transformations is useful for breaking down a function into components and visualizing the graph.
\begin{itemize}
\item $f(x)+c$ shift the graph up by $c$
\item $f(x)-c$ shift the graph down by $c$
\item $cf(x)$ stretch vertically by a factor $c$
\item $\cfrac{1}{c}f(x)$ compress vertically by a factor $c$
\item $-f(x)$ reflect the function about the $x$ axis
\item $f(-x)$ reflect the function about the $y$ axis
\item $f(x-c)$ shift to the right
\item $f(x+c)$ shift to the left
\end{itemize}
\begin{figure}[htp]
\includegraphics[width= 5in]{../images/Transforms.pdf}
\caption{Various transformations of a parabola are shown}
\end{figure}
It is useful to look at the domain and range of the basic elementary function in conjunction with the graphs of these functions. This makes things work a little easier.
We can plot the trigonometric functions. You can notice that $\sin x$ and $\cos x$ are periodic with a period of $2\pi$ they are also bounded by 1 and -1. The reciprocal functions blow up where $\sin x$ and $\cos x$ are zero. The other four functions therefore have a period of $\pi$ instead of $2\pi$. It is not that much more difficult to plot the inverse trigonometric functions. The domains of the regular functions must be restricted so they pass the horizontal line test. The usual convention can be derived from the plotted graphs.
\begin{figure}[htp]
\begin{center}
\includegraphics[width= 6in]{../images/TrigFunctions.pdf}
\caption{There are six trigonometric functions. These functions are all periodic.}
\end{center}
\end{figure}
\begin{figure}[htp]
\begin{center}
\includegraphics[width= 5in]{../images/TrigInv.pdf}
\caption{There inverse trigonometric functions can be found with some restrictions to the domain of the normal functions.}
\end{center}
\end{figure}
The hyperbolic functions differ from the trigonometric functions in that they are not periodic. Also the inverse hyperbolic functions are plotted indicating how the domain has to be restricted for $\cosh x $ and $\sinh x$ because the horizontal line test is violated.
\begin{figure}[htp]
\begin{center}
\includegraphics[width= 4in]{../images/Hyperbolics.pdf}
\end{center}
\caption{Graphs of the six hyperbolics functions near the origin}
\end{figure}
\begin{figure}[htp]
\begin{center}
\includegraphics[width= 4in]{../images/hyperinv.pdf}
\end{center}
\caption{Graphs of the six inverse hyperbolics functions near the origin}
\end{figure}
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