Mostly we've done algebra so far, with a little calculus thrown
in, and reminded ourselves that there are some equations that we can
solve to give a unique answer, and some that we can't.
We've introduced some important compositions, which is
really crucial: if we define $y=f(x)$, and $f$ is invertible, then
the compositions
\[
f^{-1}(f(x))=x
\]
and
\[
f(f^{-1}(y))=y
\]
are two identity maps -- they take a thing, and return the same identical thing.
Why do I use different variables $x$ and $y$ for these two? Is it essential?
The key to invertibility is the concept of a one-to-one function:
Is the function that maps students to seats in a typical
classroom one-to-one?
Is the function that maps dorm students at NKU to dorm
rooms one-to-one?
But we can't do any calculus with those functions....:(
Let's get into 6.2: we're continuing our review Calc I, using these new
and important functions.