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\centerline{MATH 133 -- Calculus with Fundamentals 1}
\centerline{Practice on Compositions of Functions}
\centerline{September 8, 2017}
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\noindent
{\it Background}
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Function composition is basically a fancy name for ``plugging one function
into another function.'' The slightly tricky thing to remember is that there
are two ways to do this with two functions:
\begin{enumerate}
\item $f \circ g$ means ``plug $g(x)$ into $f$'':  $(f\circ g)(x) = f(g(x))$,
\item $g \circ f$ means ``plug $f(x)$ into $g$'':  $(g\circ f)(x) = g(f(x))$.
\end{enumerate}
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\noindent
{\it Questions}

\begin{enumerate}
\item[1)]  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[2)]  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[3)] 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 2?
\end{enumerate}
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