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Here we will explore some properties of inverse functions.
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    \item The function $f(x) = 4x-3$ is invertible. Find the inverse function $f^{-1}(x)$ and plot a graph of both $f$ and $f^{-1}$. 
    \item The function $f(x) = x^2$ is not invertible. Try to solve for the inverse anyway, and sketch graphs of $f$ and everything you get by trying to solve for the inverse.
    \item What do we notice about these graphs? In particular, what is strange about the graph(s) for $f^{-1}$ when $f$ was not invertible? Think about a test we have for determining if a graph is represented by a function.
    \item Look at the graphs you have above and see if you can find a relation between the graph of $f$ and the graph of $f^{-1}$. From a more algebraic perspective, if the point $(a,b)$ is on the graph of $f$, what point has to be on the graph of $f^{-1}$? What does this mean in terms of the geometry of the graphs?
    \item With this relation you described in the previous part, how does the function test for $f^{-1}$ relate to a different type of test on the graph of $f$? That is, is there a test we could perform on the graph of $f$ to see if it will be invertible?
    \item We can fix the problem of $x^2$ being non-invertible by changing the domain. Draw a graph of $f(x) = x^2$ on $x \geq 0$. Does it pass the new test? If so, what is the inverse function to this restricted $f$?
\end{enumerate}

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