Exercise 1.4.24

Question

Compare the functions $f(x)=x^{5}$ and $g(x)=5^{x}$ by graphing both functions in several viewing rectangles. Find all points of intersection of the graphs correct to one decimal place. Which function grows more rapidly when $x$ is large?

Khagan Gulusoy · Accepted Answer

We look at the graphs of the functions $f$ and $g$ in three different viewing triangles:
\begin{figure}[H]\centering
\begin{tikzpicture}
\begin{axis}[
    axis lines=middle,
    xlabel={$x$},
    ylabel={$y$},
    domain=-5:5,
    samples=100,
]
\addplot[mark=none, color=blue, ultra thick] {x^5};
\addlegendentry{$x^5$};
\addplot[mark=none, color=red, ultra thick] {5^x};
\addlegendentry{$5^x$};
\end{axis}
\end{tikzpicture}

\begin{tikzpicture}
\begin{axis}[
    axis lines=middle,
    xlabel={$x$},
    ylabel={$y$},
    domain=0:2,
    samples=100,
]
\addplot[mark=none, color=blue, ultra thick] {x^5};
\addlegendentry{$x^5$};
\addplot[mark=none, color=red, ultra thick] {5^x};
\addlegendentry{$5^x$};
\addplot[mark=*,color=black] (1.765,17.125) node[left] {(1.765, 17.125)};
\end{axis}
\end{tikzpicture}
\begin{tikzpicture}
\begin{axis}[
    axis lines=middle,
    xlabel={$x$},
    ylabel={$y$},
    domain=0:6,
    samples=100,
    scaled y ticks=false,
]
\addplot[mark=none, color=blue, ultra thick] {x^5};
\addlegendentry{$x^5$};
\addplot[mark=none, color=red, ultra thick] {5^x};
\addlegendentry{$5^x$};
\addplot[mark=*,color=black] (5,3125) node[left] {(5, 3125)};
\end{axis}
\end{tikzpicture}
\end{figure}
The points of intersection of these graphs are (approximately) points $(1.8,17.1)$ and $(5,3125)$. The function $f(x)=3^{x}$ grows more rapidly when $x$ gets larger.

Exercise 1.4.24

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Exercise 1.4.24

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