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\begin{document}

\begin{center}
{\Large{\color{blue}Quiz 3}}
\end{center}
{\bf Instructions:} 
Show your work and explain every step. 


\begin{enumerate}

\item (4 points) Prove by a counterexamples that the following function is not onto:

\begin{center}
$g: \text{ } \R \longrightarrow \R$, $g(x)= e^{x^2}$.
\end{center}

\vspace{.5 in}

\item (4 points) Prove by a counterexamples that the following function is not one-to-one:

\begin{center}
$f: \text{ } \R \longrightarrow \R$, $f(x)= e^{(x-3)^2}$.
\end{center}

\vspace{.5 in}


\item (5 points) Find the inverse of the following function:

\begin{center}
$h: \text{ } \R \longrightarrow (0,\infty)$, 
$h(x)=e^{x-2}$.
\end{center}

\vspace{1.5 in}

\item (3 points) Let 
$f: \text{ } \R \longrightarrow \R$, $f(x)= x-3$ and let $g$ be the function in Question (1). Find $g \circ f$ if it's 
defined.

\vspace{.5 in}


\item (4 points) Determine if the following sets are countable or not. Explain.

\begin{enumerate}
\item $5\N$.

\item $\{ \frac{1}{2n} ~|~ n \in \N\}$.

\item $(3,8)$.

\item $(3, \infty)$.

\end{enumerate}

\end{enumerate}


\end{document}