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\begin{document}

\begin{center}
CSIT 241 - EXAM II
\end{center}

{\bf Name: } 

{\bf Instructions:} Do all of the following. EXPLAIN every step. Points 
will be deducted for incomplete proofs or incomplete solutions. Do not use calculators. Do not look at your 
neighbor or talk to him. If you use a method different than what the question is asking for, you'll get no credit. 
Use only notation used in class. Any violation of the instructions may result in a partial credit or no credit at all. 

The exam is closed book, closed notes. 
No material may be used and the Internet is not allowed.

\vspace{.2 in}

{\bf Time:} 50 minues. 

\begin{enumerate}

\item (3 points) Prove that the function $f: \text{ }\R^+ \longrightarrow (1,e^2)$, 
defined by $f(x)=e^{(\frac{2x}{x+3})}$ is one-to-one.

\vspace{2.5 in}

\item (3 points) Prove by a counterexample that the function $f: \text{ }\R \longrightarrow \R$, 
defined by $f(x)= -5 + e^{(2x-7)^2}$ is not one-to-one.

\vspace{.5 in}


\item (3 points) Prove by a counterexample that the function $f: \text{ }\R \longrightarrow \R$, 
defined by $f(x)= -5 + e^{(2x-7)^2}$ is not onto.

\vspace{1 in}

\item (3 points) Let $f: \text{ }\R \longrightarrow \R^+$, 
defined by $f(x)=e^{2x+7}$ and $g: \text{ }\R \longrightarrow \R$, 
defined by $g(x)=x^2 + 8$ . Find $g \circ f(x)$ if defined.


\newpage

\item (5 points)Let $f: \text{ } (\frac{5}{2}, \infty) \longrightarrow \R$, 
defined by $f(x)= \ln(2x-5)$. Find $f^{-1}(x)$.

\vspace{3.5 in}

\item (3 points) Prove that the set $S = \{\frac{(-1)^n}{5n} ~|~ n \in \N\}$ is countable.

\newpage 

\item (24 points) Deteemine if the following are true or fasle

\begin{enumerate}

\item If $g \circ f$ is a bijection, then both $f$ and $g$ are bijections.

\item If $g$ and $f$ are both invertible, then $g \circ f$ is invertible, and $(g \circ f)^{-1}$ is invertible, 
and $(g \circ f)^{-1} = f^{-1} \circ g^{-1}$.

\item The interval $(7,9)$ is countable.

\item If $a$ is a non zero real numbers, then $\gcd(a,a) = a$ and $gcd(-a,a) = a$.

\item Let $n$ be a positive integer greater than 1. If all prime numbers less than $\sqrt{n}$ don't divide $n$, then 
$n$ is prime.

\item If $A$ is a square matrix with a repeated column, then $\det(A) = 0$.

\item If $A$ is an $n \times n$ matrix and $x$ and $b$ are $n \times 1$ vectors, then the 
system $Ax=b$ has a unique solution.

\item If $A$ and $B$ are matrices such that $AB=0$, then either $A=0$ or $B=0$.

\item If $a$ and $b$ are nonzero integers, then $\lcm(a,b) = \lcm(b,-a)$.

\item If $a$ and $b$ are vectors such that $a \cdot b = 0$, then either $a=0$ or $b=0$.

\item If $f$ is a function from $A$ to $B$, $X \subseteq A$, and $Y \subseteq B$, then 
the image of the preimage of $Y$ is equal to $Y$, and the preimage of the image of $X$ is equal to $X$.

\item The union of a countable set and an uncountable set is uncountable. 

\end{enumerate}


\newpage


\newpage

\item 

\begin{enumerate}

\item (8 points) Find $\gcd(100,41)$ and write it as a linear combination of 100 and 41. Find also $\lcm(41,-100)$.

\vspace{2.5 in}

\item (3 points) Find the canonical factorization of $20000$.

\vspace{1 in}

\item (3 points) Find $41^{-1}$ in $\Z_{100}$.

\vspace{.5 in}

\item (3 points) Find $200 \oplus 50$ in $\Z_{100}$.

\vspace{.5 in}

\item (3 points) Solve the equation $400 x = 1$ in $\Z_{10000}$.

\vspace{.5 in}

\item (3 points) Solve the equation $4 x = 0$ in $\Z_{10}$.

\newpage

\end{enumerate}

\item Let \[A =
\left[
\begin{array}{ccc}
1 & 0 & 3 \\
0 & 5  & 4\\
3 & 2 & 0
\end{array}
\right].\] 

\vspace{.3 in}

\begin{enumerate}

\item (7 points) Find $(2A-3I)^T$.

\vspace{3 in}

\item (8 points) Find $A^2$.

\vspace{3 in}


\newpage

\item (6 points) Find $\det(A)$.

\vspace{1.5 in}

\item (8 points) Find $A^{-1}$.

\vspace{4 in}

\item (4 points) Use $A^{-1}$ to find the solution of the system (do not use substitution or elimination):

$x_1 + 2 x_2 + 3 x_3 = -1$.

$4 x_1 + 5 x_2 + 7 x_3 = -2$.

$3 x_1 + 6 x_2  = -3$.

\end{enumerate}



\end{enumerate}

\end{document}

