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\centerline{MATH 135 – Calculus 1}
\centerline{Practice on Inverse Functions}
\centerline{September 9, 2016}
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\noindent
{\it Background}
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If a function $f$ from a domain $D$ to a range $R$ has the one-to-one property (that is, each $y$ in
the range $R$ is $y = f(x)$ for exactly one $x$ in the domain $D$), then we can define a second function
$f^{-1}$ with domain $R$ and range $D$ such that
$$x = f^{-1}(y)\ {\rm exactly\ when}\  y = f(x).$$
This is called the inverse function of $f$. As we said in today’s video: CAUTION This is not the
same as $\frac{1}{f}$ (!) We want to work with a few examples of this idea today.
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\noindent
{\it Questions}
\begin{enumerate}
\item[1)] Look at Figure 19 on page 39 of our text.
\begin{enumerate}
\item[(a)] Which of the graphs there define functions with the one-to-one property?
\item[(b)] For the ones that do not have that property, 
how could we restrict the domain to get an inverse function?
Is there just one way to do that, or is there more than one?
\item[(c)] Recall from Video 1.5 that there is a geometric way to get the graph of an inverse
function from the graph of a one-to-one function. Draw the graphs of inverse functions
for each of the ones from part (a) and then for each restricted domain function from
part (b).
\end{enumerate}
\item[2)] For this problem explain why the given function has the one-to-one property. 
Then find a formula for
the inverse function by setting up an equation $y = f(x)$ and solving for $x$ as a function of $y$.
As a last step, swap the variables to write the inverse function as a function of $x$\footnote{As some of you may know, you can also swap the variables first then solve for $y$ as a function of $x$ -- you'll
get to the same place in the end.  If you're more comfortable doing it that way, by all means 
do.}.
In each case state what the domain of the inverse function is.
\begin{enumerate}
\item[(a)] $f(x) = 7x + 3$
\item[(b)] $f(x) = \displaystyle{\frac{1}{x+1}}$
\item[(c)] $f(x) = x^2 + 4x + 5$ on the domain $D = [-2,+\infty)$.
\end{enumerate}

\end{enumerate}
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