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\begin{center}\textbf{Problem Set 7 \hspace{.7in} Foundations \hspace{.7in} Due 3/21/22} \end{center}

\noindent Four of these problems will be graded (my choice, not yours!), with each problem worth 5 points.  Clear and complete justification is required for full credit.  You are welcome to discuss these problems with anyone and everyone, but must write up your own final submission without reference to any sources other than the textbook and instructor.  Submit as a pdf on Moodle or on paper.

\begin{enumerate}
\item If $f: \mathbb{R} \rightarrow \mathbb{R}$ and $g: \mathbb{R} \rightarrow \mathbb{R}$ are surjective functions, then $f + g$ is surjective.

\item If $f: A \rightarrow B$ and $g: B \rightarrow C$ are injective functions, then $g \circ f$ is injective.

\item If $f: A \rightarrow B$ and $g: B \rightarrow C$ are surjective functions, then $g \circ f$ is surjective.

\item If $f: A \rightarrow B$ has an inverse function $g$, then $g$ has $f$ as an inverse function also.

\item Let $f: A \rightarrow B$ be a bijective function.  Then there exists an inverse function $g$ for $f$.

\item Let $f: A \rightarrow B$ be a bijective function.  Then the inverse function of $f$ is unique, i.e. if $g_1$ and $g_2$ are both inverse functions for $f$, then $g_1 = g_2$.

\item Let $f: A \rightarrow B$ be an invertible function.  Then $f$ is bijective.

\item	Do the InverseFunctions assignment on WeBWorK, available via
\begin{center} http://http://crlinwebwork2.coe.edu/webwork2/MTH-215/InverseFunctions/ \end{center}

\item Explain clearly the domain issue with your answer to the last problem on the InverseFunctions WeBWorK assignment.

\item Critique the following proof of the proposition that all functions are bounded:
Take $x \in \mathbb{R}$.  Then since $|f(x)|+1>|f(x)|$, then we can take $M = |f(x)|+1$ as a bound for $f$.

\end{enumerate}

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