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\markright{Chapter 1: Answers 2}
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\Bf{Chapter 1: Answers 2 \hfill Jack K. Cohen \hfill Colorado School of Mines}

\begin{enumerate}
\item
        \begin{enumerate}
        \item   $y = x + 5$: function; domain and range are all real numbers.
        \item   $y = \sin x + \cos x$: function; domain is all real numbers, but the range is hard to pin down exactly (without some calculus).  From a graph, it is roughly $(-1.4, 1.4)$.
        \item   $y = 1 / x$: function, domain is $x \neq 0$ and range is all real numbers except 0.
        \item   $y - x - 5 = 0$: still a function; domain and range are all reals.
        \item   $x^2 + y^2 - 4 = 0$: \Em{not} a function---for example at $x = 0$ there are two $y$-values.
        \item   $x^2 + y^2 + 4 = 0$:  No points satisfy this equation since the left side is greater than 4.  Only a pedant would care if it should be called a function with an empty domain or if it shouldn't be accepted as a function at all.
        \end{enumerate}

\item Functions can sometimes be specified by tables.
        \begin{enumerate}
        \item The ``Denver Diary'' for July 15, 1991 certainly specifies Temp as a function of Time with the domain being the times from midnight to 8pm the following day (the paper went to press before the temperature at 9pm was available) and the range being the temperatures that appear in the table.  On \Em{some} days Time might be a function of hourly temperature, but on July 15, 1991, it so happened that the temperature 96 degrees occured at both 4pm and 6pm, so it was \Em{not} a function of time on that particular day (in common sense terms, given the temperature of 96 degrees, we could not decide if the corresponding time was 4pm or 6pm).
        
        \item The ``Never on Sunday'' data set specifies Answer as a function of Day with the domain being the 7 days and the range being the set of answers, \{``yes'', ``no''\}.  But Day is certainly not a function of Answer, since a ``yes'' leaves us with $6$ possible choices for the correct day.
        \end{enumerate}

\item
        \begin{enumerate}
        \item  Yes, domain and range are the set \{1, 2, 3, 4, 5\}.
        \item No.  There are 2 values corresponding to the value 1 in the domain.
        \end{enumerate}

\item  Graph ``a'' is a function.  Its domain and range can only be determined approximately from the graph.  This graph was actually created by evaluating the function $\sin(\sin(\sin(\sin(x))))$ on the set (the domain!) of points, $-2\pi + k \pi/16, k = 0, 1, \ldots, 64$.  Check it out, if you have the patience.

\noindent  Graph ``b'' does \Em{not} represent a function (two values for the positive $x$ values).  It happens to be the graph of $y^2 = x$.

\item
        \begin{enumerate}
        \item Yes.
        \item Yes.
        \item Yes, but no one has the precise data.
        \end{enumerate}

\item
        \begin{enumerate}
        \item If $x_1 \neq x_2$, the formula is: $y - y_1 = m (x - x_1)$, where $m = (y_2 - y_1)/(x_2 - x_1)$.   If $x_1= x_2$, the formula is: $x = x_1$.
        
        \item Once again: $y - y_1 = m (x - x_1)$, but now $m$ is given directly.
        
        \item $y = m x + b$
        \end{enumerate}


\end{enumerate}
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