<html><head><meta name="color-scheme" content="light dark"></head><body><pre style="word-wrap: break-word; white-space: pre-wrap;">\sethandouttitle{The Exponential and Logarithm Functions}


\begin{enumerate}

	\item	Evaluate $f(x) = x^2$ and $g(x) = 2^x$ at $x=-3,-2,-1,0,1,2,3,4,5$. Graph both functions on the same graph.

	\item   Find the domain of $\displaystyle f(x) = \frac{ 1 - e^{x^2} }{ 1 - e^{1-x^2} }$ and $\displaystyle g(x) = \frac{ 1+x }{ e^{\cos x} }$.

	\item	Find the limit.
		\begin{enumerate}
			\item $\displaystyle\lim_{ x \rightarrow \infty } (1.0001)^x $.
			\item $\displaystyle\lim_{x \rightarrow \infty} \frac{ e^{3x} - e^{-3x} }{ e^{3x} + e^{-3x} }$.
			\item $\displaystyle \lim_{x \rightarrow \infty} ( e^{-2x} \cos x)$.
		\end{enumerate}

	\item 	Is $g(x) = 1/x$ one-to-one?

	\item 	If $g(x) = 3 +x + e^x$, find $g^{-1}(4)$.

	\newpage

	\item	The formula $C = \frac{5}{9}(F-32)$ expresses Celsius $C$ as a function of Fahrenheit $F$. 
			Find a formula for the inverse function and interpret it.

	\item 	Find the inverse of the function $f(x) = 1 + \sqrt{ 2 + 3x }$.

	\item 	Find the inverse of the function $f(x) = e^{2x-1}$.

	\item 	Find $ (f^{-1})'(a)$ using the inverse derivative formula.
		\begin{enumerate}
			\item $f(x) = x^3$.
			\item $f(x) = 9-x^2$.
		\end{enumerate}
		Then, find the inverse function first and differentiate it. Do your answers agree?


	\item 	Find the exact value of each expression (without a calculator).
		\begin{enumerate}
			\item	$\log_5(125)$.
			\item 	$\displaystyle \log_3\left( \frac{1}{27} \right)$.
			\item	$\log_2 6 - \log_2 15 + \log_2 20$.
		\end{enumerate}
		
	\item	Solve for $x$:
		\begin{enumerate}
			\item $e^{7-4x}=6$.
			\item $\ln(3x-10) = 2$.
			\item $\ln x + \ln (x-1) = 1$.
		\end{enumerate}

\end{enumerate}   



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