Exercise 1.4.3

Question

\begin{enumerate}[label=(\alph*)] \item Write an equation that defines the exponential function with base $b>0$. \item What is the domain of this function? \item If $b eq 1$, what is the range of this function? \item Sketch the general shape of the graph of the exponential function for each of the following cases. \begin{enumerate} \item [(i)] $b>1$ \item [(ii)] $b=1$ \item [(iii)] $0

Khagan Gulusoy · Accepted Answer

\begin{enumerate}[label=(\alph*)]
\item $$f: \mathbb{R} 	o \mathbb{R}_{+}, \quad f(x) =b^{x},$$

where $b$ is some positive real number.

\item  This function is defined for all real numbers.

\item  If $b 
eq 1$, the range of this function, as noted in (a), is the set of all positive real number y $\mathbb{R}_{+}$.

\item \begin{enumerate}[label=(oman*)]
\item For $b>1,$ we have:
\begin{figure}[H]\centering
\begin{tikzpicture}
\begin{axis}[axis lines=middle,
    xlabel={$x$},
    ylabel={$y$},
    domain=-2:2, % Adjust the domain as needed
    samples=200, % Adjust the number of samples for a smoother plot
    legend style={
        at={(0.95,0.95)},
        anchor=north east,
    },
]
\addplot[color=blue, ultra thick] {exp(x)};
\end{axis}
\end{tikzpicture}
\end{figure}

\item For $b=1$, we have:
\begin{figure}[H]
\begin{tikzpicture}
\begin{axis}[axis lines=middle,
    xlabel={$x$},
    ylabel={$y$},
    domain=-2:2, % Adjust the domain as needed
    samples=200, % Adjust the number of samples for a smoother plot
    legend style={
        at={(0.95,0.95)},
        anchor=north east,
    },
]
\addplot[color=blue, ultra thick] {1};
\end{axis}
\end{tikzpicture}
\end{figure}

\item For $0<b<1$, we have:
\begin{figure}[H]
\begin{tikzpicture}
\begin{axis}[axis lines=middle,
    xlabel={$x$},
    ylabel={$y$},
    domain=-2:2, % Adjust the domain as needed
    samples=200, % Adjust the number of samples for a smoother plot
]
\addplot[color=blue, ultra thick] {0.5*exp(x)};
\end{axis}
\end{tikzpicture}
\end{figure}
\end{enumerate}

\end{enumerate}

Exercise 1.4.3

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Exercise 1.4.3

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