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\begin{document}
	
	\mbox{}
	\title{Problem Set 1: Elementary Functions}
	
	\maketitle
	
	
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%%%%%%%%%%%%%%%%%%%%%%%%%%%%
	%\section{Introduction} \label{sec:intro}
	\noindent {\footnotesize Note: Most of the problems were taken from the textbook} \cite{jS15}.
	
	\begin{problem}
		Find the domain of the functions:
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $g(x) = \sqrt{3-t} - \sqrt{2-t}$;
			\item $f(u) = \frac{u+1}{1 + \frac{1}{u+1}}$;
			\item $h(x) = \frac{\cos x}{1 - \sin x}$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Find the domain and sketch the graph of the functions:
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $f(x) = \sqrt{4-x^2}$;
			\item $h(t) = \frac{t^2 - 1}{t+1}$;
			\item $f(x) = \twopf{x+1}{x \le -1}{x^2}{x > -1}$;
			\item $y = \big{|} |x| - 1\big{|}$.
		\end{enumerate}
	\end{problem}

	\begin{problem}
		Express the area of an equilateral triangle as a function of the length of a side.
	\end{problem}
	
	\begin{problem}
		Classify each function as a power function, root function, polynomial (state its degree), rational function, algebraic function, trigonometric function, exponential function, or logarithmic function.
		\begin{enumerate}[label=\alph*), topsep=8pt, itemsep=8pt, partopsep=1ex, parsep=1ex]
			\item $f(x) = \log_2 x$;
			
			\item $y = x^2(2 - x^3)$;
			
			\item $g(\theta) = \tan \theta - \cos^2 \theta$;
			
			\item $y = \pi^x$;
			
			\item $y = \frac{\sqrt{x^3 - 1}}{1 + \sqrt[3]{x}}$;
			
			\item $h(z) = x^z$;
			
			\item $y = \frac{s}{e^2 + s}$.
		\end{enumerate}
	\end{problem}
	
	\begin{problem}
		What do all the members of the family of linear functions $f(x) = 1 + m(x+3)$ have in common? Sketch several members of the family.
	\end{problem}
	
	\begin{problem}
		Find an expression for a cubic function $f$ if
		\[
			f(1) = 6 \quad  \text{and} \quad f(-1) = f(0) = f(2) = 0.
		\]
	\end{problem}
	
	\vspace{20pt}
	
	\begin{thebibliography}{20}
		
		\bibitem{jS15} J.~Stewart: \emph{Single Variable Calculus} 8th Edition, Cengage Learning, Boston 2015.
		
	\end{thebibliography}
	
\end{document}