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Problems PreCal 1508 PLTL Workshop September 7, 2011\\
PL: Alex Knaust, Lecturer: Yi-Yu Liao
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\begin{enumerate}
		\item Write the equation of the function $p(x)$ shifted up by $k$ units and to the left by $h$ units.
	
		\item Write down the definition for a function with domain $A$ and range $B$. And give a real life example of a function (without numbers )
	
		\item Determine $(f + g)(x)$, $(f \cdot g)(x)$, $(\frac{f}{g})(x)$, and $(f \circ g)(x)$ and determine the domain of the resulting function, for the following functions
		\begin{enumerate}
			\item $g(x) = x^2 + 3,\quad f(x) = x^3 + 9 $
			\item $g(x) = 3x^2 + 1,\quad f(x) = \sqrt{1 - x}$
			\item $g(y) = \sqrt{y^2 - 4}, \quad f(x) = \frac{x^2}{x^2 + 1}$
			\item $f(s) = \frac{1}{s}, \quad g(y) = \frac{1}{y^2}$
			\item $f(r) = \frac{r}{r + 1}, \quad g(q) = q^3$
		\end{enumerate}
				\item Show that the product of two odd functions is an even function, and that the product of two even functions is an even function
		\item Find two different pairs of functions $f$ and $g$ such that $(f \circ g)(x) = h(x)$. There are many correct answers.
		\begin{enumerate}
			\item $h(x) = (2x + 1)^2$
			\item $h(x) = \frac{-x^2 + 3}{4 - x^2}$
			\item $h(x) = \frac{4}{(5x + 2)^2}$
			\item $h(x) = \sqrt[3]{9 - x}$
		\end{enumerate}
		

		\item Consider $f(x) = x^2$ and $g(x) = \sqrt{x}$. Find  $f \circ g$ and $g \circ f$, and their domains.
		
		\item Find $(f \circ f)(x)$ and the implied domain of $f \circ f$ if $f(x) = \sqrt{-x}$
				
		\item Graph $f(x) = -|-x + 2| + 1$
		
		\item Expand $(x+2)^3 + (x - 2)^3$

\end{enumerate}
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