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\begin{document}

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\section*{Written Assignment 6}

\begin{question}{1}
Define the following terms:
\begin{itemize}
\item A function
\item An injective function
\item A surjective function
\item A bijective function
\end{itemize}
\end{question}

\begin{question}{2}
Give examples of:
\begin{itemize}
\item A relation which is \textbf{not} a function.
\item A function which is \textbf{not} injective. Specify the domain and codomain.
\item A function which is \textbf{not} surjective. Specify the domain and codomain.
\end{itemize}
\end{question}

\begin{question}{3}
Consider the function:
$$f(n) = 3n \text{ for } f : \mathbb{Z} \rightarrow \mathbb{Z} $$
\begin{enumerate}[label=(\alph*)]
\item What is this function's image?\\
\textbf{Hint:} Use set-builder notation.
\item Is this function injective? Explain.
\item Is this function surjective? Explain.
\item If this function is injective, what is its inverse?
\end{enumerate}
\end{question}

\begin{question}{4}
Expand the function $f \circ f$, with $f$ as defined in the previous problem. Specify the domain and codomain.
\end{question}

\begin{question}{5}
Consider the function:
$$g(x) = \frac{1}{2} - \frac{1}{4} x^2 \text{ for } g : \mathbb{R} \rightarrow \mathbb{R}$$
\begin{enumerate}[label=(\alph*)]
\item What is this function's image?\\
\textbf{Hint:} Graph the function for a visualization.
\item Is this function injective? Explain.
\item Is this function surjective? Explain.
\item If this function is injective, what is its inverse?
\end{enumerate}
\end{question}

\clearpage

\begin{question}{6}
Consider the function:
$$h(x) = \frac{1-2x}{4-x} \text{ for } h : ? \rightarrow \mathbb{R}$$
\begin{enumerate}[label=(\alph*)]
\item For what domain is this function defined?
\item What is this function's image?
\item Is this function injective? Explain.
\item Is this function surjective? Explain.
\item If this function is injective, what is its inverse?
\end{enumerate}
\end{question}

\end{document}
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