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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Practice on Combinations of Functions}
\centerline{September 7, 2017}
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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. 
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\noindent
{\it Questions}
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\begin{enumerate}
\item[1)]  Give an 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 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)]  $f(x) = \frac{1}{x^2 + 1}$, $g(x) = \cos(x)$
\item[(c)]  $f(x) = x + \frac{1}{x}$, $g(x) = f(x)$.
\end{enumerate}
\item[3)]  Express each of the following functions $F(x)$ as $F = f\circ g$ for some 
functions $f,g$:
\begin{enumerate}
\item[(a)] $F(x) = \sqrt{1 - x^2}$
\item[(b)] $F(x) = 2^{x^2}$
\item[(c)] $F(x) = \frac{1}{x^2 + 3x + 2}$
\end{enumerate}
\item[4)] Recall our ``rule of thumb'' on domains:  If $f(x)$ is given by a formula, then
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?
\end{enumerate}
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