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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Practice on Functions, Graphs, Shifting and Scaling}
\centerline{September 7, 2015}
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\noindent
{\it Background}
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Recall from the video ``lecture'' for last Friday that a {\it function} $f$ from a {\it domain} $D$ to a set $Y$
is a rule that assigns to each element $x$ in $D$, a unique element $f(x)$ in $Y$.  This means that 
the graph of $f$ passes the ``vertical line test:'' for each $x$ in $D$, there is exactly one 
point $(x,f(x))$ on the graph of $f$.  The set of values $f(x)$ for $x$ in $D$ is called the {\it range}
of the function.  The range is a subset of $Y$ (it can be all of $Y$ but it does not have to be).  
(Almost) all of the functions we will see have subsets of the real numbers as their domains and ranges.
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\noindent
{\it Questions}
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\begin{enumerate}
\item[1)]  The question refers to the 
plot of temperature versus time over a 24-hour period at a particular location
that we saw Friday (see back of this page). Is this the graph of a function?  
If so, what are the domain and range, as intervals?
If not, why does it fail to be a function?
\item[2)]  Look at Figure 26 on page 10 of our text book.  Which of these graphs is the graph of a function?
Explain.
\item[3)]  If no domain is specified, for a function defined by a formula, then the ``rule of thumb''
is to take the domain to be the set of all real $x$ such that the formula gives a well-defined value.
Using this, 
\begin{enumerate}
\item[(a)]  What is the domain of the function defined by $f(x) = \frac{1}{x^2 - 4}$?
\item[(b)]  What is the domain of the function defined by $f(x) = \sqrt{4 + x}$?
\end{enumerate}
\item[4)]  All parts of this question deal with $f(x) = x^3$.
\begin{enumerate}
\item[(a)]  Sketch the graph $y = f(x)$ on the domain $[-2,2]$
\item[(b)]  Sketch the graph $y = f(x) - 3$ on the same domain as in part (a).
\item[(c)]  Sketch the graph $y = f(x + 1)$ on the domain $[-3,1]$ using your graph in part (a)
\item[(d)]  Sketch the graph $y = 2 f(x)$ on the domain $[-2,2]$.
\item[(e)]  Sketch the graph $y = f(2x)$ on the domain $[-1,1]$.  How is this graph related to $y = f(x)$?
\item[(f)]  Sketch the graph $y = f\left(\frac{x}{2}\right)$ on the domain $[-4,4]$.  How is this graph 
related to $y = f(x)$?
\item[(g)]  (Practice on finding patterns)  
From section 1.1 of the text and videos, we know $y = cf(x)$ is a vertical stretching or compression of $y = f(x)$.  What is the 
corresponding description of the graph $y = f(cx)$?  How does this depend on the size of $c$?
(Hint: Try drawing the graphs $y = f(x) = x^3$ on $[-2,2]$ and the graphs from part (e) and part (f) with 
those given domains, together on one set of axes.  Note that you'll need to take $x$ in the interval 
$[-4,4]$ to get all of them to ``fit'' but draw the graphs exactly as you did before, i.e. on the domains
as stated before.)
\end{enumerate} 
\end{enumerate}
\begin{figure}
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\caption{Figure for Question 1}
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