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Problems PreCal 1508 PLTL Workshop, September 28, 2011\\
PL: Alex Knaust, Lecturer: Yi-Yu Liao\\

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Please do the problems that you feel will help your group the most first (you don't have to do them in order). I suggest starting with \#1 as it requires you to think critically about the properties of rational functions.
\begin{enumerate}
\item Create a rational function, $q(x)$ with the following properties (there are many correct answers).
\begin{enumerate}
\item $q(x)$ has a hole at $x = 1$.
\item $q(x)$ has two vertical asymptotes $x = - 1,\quad x = 3$.
\item $q(x)$ has a horizontal asymptote at $y = 0$.
\item $q(x)$ has $(2, 0)$ as an $x$-intercept.
\end{enumerate}

\item Find all vertical, horizontal and slant asymptotes of the functions, as well as any x-intercepts or y-intercepts and holes. Use this to graph the function...
\begin{enumerate}
\item $f(x) = \frac{x^2}{x^2 + 9}$
\item $f(x) = \frac{x^2 - x - 2}{x^3 - 2x^2 -5x + 6}$
\item $h(t) = \frac{2t^3 + t^2 - 8t - 4}{t^2 - 3t + 2}$
\item $f(x) = 2^{-2x}$
\item $g(x) = 3^{x}$
\end{enumerate}

\item Determine which values of $x$ satisfy the equation.
\begin{enumerate}
\item $e^{x - 9} = e^{8}$
\item $2^{x^2 - 1} = 1$
\item $\frac{1}{32} = 2^{5x}$
\item $\left(e^{x - 3}\right)^2 = e^{-9}$
\end{enumerate}

\item Are the following true or false? Explain your reasoning/give a counterexample.
\begin{enumerate}
\item Any $a^x$ can be written as a product of $a$. i.e. $\underbrace{a\cdot a\cdot a\cdots a}_x$
\item The functions $f(x) = \frac{x^2 + 3x + 2}{x+2}$ and $g(x) = x + 1$ are equal (as functions).
\item The range of $e^{-2392x}$ is all real numbers.
\item Exponential functions have inverse functions
\item If $f(x)$ is a rational function with domain $(-\infty, a)\cup(a, \infty)$ Then the graph of $f(x)$ has a hole at $a$.
\item You can use the one-to-one property of exponential functions to solve $2^{2x - 1} = 3^{3x +1}$
\end{enumerate}

\item Graph the inverse function of $e^x$ by reflecting it across $y=x$. What is its domain and range?

\item Which will "grow" faster eventually, $2^x$ or $x^2$?

\end{enumerate}
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