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\begin{document}

\begin{center}
{\Large{\color{red}Assignment 4}}

{\it Due Friday, March 20, 07 at 11:00 AM in Class}
\end{center}
{\bf Remarks:} 
I may not grade all assignments and may not grade all questions/parts on the assignment I decide to grade. 
You're welcome to ask me for help. 
Show your work and explain every step. 


\begin{enumerate}

\item Let $f: \text{ } \R-\{2\} \longrightarrow \R^+$, 
$f(x)=e^{(\frac{3x-7}{x-2})}$.

\begin{enumerate}

\item Is $f$ well-defined? Explain.

\item If the answer to the first part is yes, is $f$ onto? Explain.

\item If the answer to the first part is yes, is $f$ one-to-one? Explain.

\end{enumerate}


\item Prove the following functions are bijections by proving they are one-to-one and onto. Also, find 
the inverse of each function:

\begin{enumerate}

\item $f:~ (3,\infty) \longrightarrow \R$, $f(x) = 2 + \frac{1}{2}\text{ln}(x - 3)$.

\item $f: \text{ } \R-\{2\} \longrightarrow \R - \{3\}$, 
$f(x)=\frac{3x-7}{x-2}$.

\end{enumerate}

\item Prove by counterexamples that the following functions are not one-to-one and not onto:

\begin{enumerate}
\item $f:~\R \longrightarrow (-66.5,\infty)$, $f(x) = 9(x - 777)^{8} - 66$.

\item $f: \text{ } \{1,2,3,...,n\} \longrightarrow \Z$, $f(k)=\frac{n!}{k!(n-k)!}$, where $n$ is an integer greater 
than 10. (Do not substitute any value for $n$. Thus, in your counterexample, the two values in the domain that have 
the same image must be in terms of $n$.)
\end{enumerate}

\item Decide if $g \circ f$ is defined in each of the folklowing cases. If it's defined, find it:

\begin{enumerate}

\item $f:~\R \longrightarrow [3,\infty)$, $f(x) = 2 x^2 + 3$, $g:~\R \longrightarrow \R$, $g(x) = 4 x - 9$.

\item $g:~\R \longrightarrow [3,\infty)$, $g(x) = 2 x^2 + 3$, $f:~\R \longrightarrow \R$, $f(x) = 4 x - 9$.

\item $g:~\R^+ \longrightarrow \R$, $g(x) = \text{ln}(x)$, $f:~\R \longrightarrow \R$, $f(x) = 4 x - 9$.
\end{enumerate}

\item Prove that $|5 \Z + 1|=|10 \Z + 8|$ by finding a bijection between the two sets.

\end{enumerate}

\end{document}