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\centerline{MATH 135 -- Calculus 1}
\centerline{Practice on Combinations of Functions, Piecewise-defined functions}
\centerline{September 5, 2016}
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\noindent
{\it Background}
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In section 1.3, various ways of combining known functions to produce new ones are 
discussed.  This also leads to some important classes of functions such as 
polynomial functions, rational functions, algebraic functions and transcendental functions, 
as well as piecewise-defined functions.
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\noindent
{\it Questions}
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\begin{enumerate}
\item[1)]  Give an \emph{example} of a function in each of the classes of functions from the 
Background above.  
\item[2)]  For each of the following pairs of functions $f,g$, compute \emph{both} compositions
$f\circ g$ and $g\circ f$.  
\begin{enumerate}
\item[(a)]  $f(x) = x^2 + 4x + 5$, $g(x) = \sqrt{x}$
\item[(b)]  $f(x) = \frac{1}{x^2 + 1}$, $g(x) = \cos(x)$
\end{enumerate}
\item[3)]  Express each of the following functions $F(x)$ as compositions $F = f\circ g$ for some 
functions $f,g$ (Note:  There is more than one correct answer here! Do you see why?)
\begin{enumerate}
\item[(a)] $F(x) = \frac{1}{\sqrt{1 - x^2}}$
\item[(b)] $F(x) = (x^2 + 1)^{4/3}$.
\end{enumerate}
\item[4)] From videos for last week, our ``rule of thumb'' on domains is:  
If $f(x)$ is given by a formula, then
\emph{unless stated otherwise} we take the domain to be the set of all real $x$ that yield a
well-defined result in the formula.  What are the domains of the functions $F$ in 
question 3?
\item[(5)] For the upcoming tax season on income earned
in 2016, for married couples filing joint 
returns, the Federal income tax due will be computed by something like the following tax
bracket table.
The \emph{AGI} is the \emph{adjusted gross income} (income minus exemptions, deductions for things
like mortage interest paid or medical care expenses, plus credits for various things, etc.\footnote{Computing this can get \emph{very} involved -- ask your parents or a tax accountant some time.})
\emph{In each case, to save space, only the {\bf upper limit} of the tax bracket is given}.  So for example the
second entry $75.3$ means all AGI values in the interval starting at the previous level
$18.55$ up to $75.3$, i.e. $[18.55,75.3)$ or $\$18550 \le AGI < \$75300$.
The \emph{tax rate} is the percentage of the AGI that is due as tax.  (Of course, in almost all
cases, most or all of this will have been withheld from paychecks through the year.)
\[
\begin{array}{lllllllll}
AGI\ (in\ \$1000) & \vline & 18.55 & 75.3 & 151.9 & 231.45 & 413.35 & 466.95 &{\rm higher} \\ \hline
tax\ rate & \vline & 10\% & 15\% & 25\% & 28\% & 33\% & 35\% & 39.6\%
\end{array}
\]
Write the \emph{tax due} as a function of the AGI using this information. 
\end{enumerate}
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