Exercise 1.3.32

Question

Use the given graph of $f$ to sketch the graph of $y = 1/f(x).$ Which features of $f$ are the most important in sketching $y = 1/f(x)$? Explain how they are used.

Khagan Gulusoy · Accepted Answer

The following properties are true for the graph of the function $1/f$:
\begin{itemize}
    \item If the graph of $f$ crosses the point $(x,0)$ where $x$ is some real number, the vertical line that represents the equation $y=x$ is the asymptote of the graph of $1/f.$ Conversely, if the values of the function $f$ become increasingly larger, the graph of $1/f$ comes closer and closer to the $x$-axis.
    \item Local maximal values of $f$ are local minima of $1/f$ on the same intervals.
    \item Local minimal values of $f$ are local maxima of $1/f$ on the same intervals.
\end{itemize}
Using the following properties, we sketch the graph of $1/f$ as follows:
    \begin{figure}[H]\centering
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, unit vector ratio={1 1}, xmin=-3, xmax=5, ymin=-4, ymax=5, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=6mm}, y label style={anchor=north, above=3mm}]
         \addplot[domain=-3:5, samples=100, mark=none, color=blue, ultra thick, restrict y to domain=-4:5] {1 - 0.137879* x - 1.00985 *x^2 + 0.133333 *x^3 + 0.00984848* x^4 + 0.00454545 *x^5};
    \end{axis}    
    \end{tikzpicture}
    
    \begin{tikzpicture}
    \begin{axis}[axis lines=middle, unit vector ratio={1 1}, xmin=-3, xmax=5, ymin=-4, ymax=5, xlabel={$x$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=6mm}, unbounded coords=jump]
         \addplot[domain=-3:4.00001, samples=100, mark=none, color=blue, ultra thick, restrict y to domain=-4:5] {1/(1 - 0.137879* x - 1.00985 *x^2 + 0.133333 *x^3 + 0.00984848* x^4 + 0.00454545 *x^5)};
         \addplot[domain=4.00001:5, samples=1000, mark=none, color=blue, ultra thick, restrict y to domain=-4:5] {1/(1 - 0.137879* x - 1.00985 *x^2 + 0.133333 *x^3 + 0.00984848* x^4 + 0.00454545 *x^5)};
    \end{axis}    
    \end{tikzpicture}
\end{figure}

Exercise 1.3.32

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Exercise 1.3.32

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