Inverse Functions
If $f$ is a function, the inverse function $f^{1}$ (if it
exists) undoes whatever $f$ does. That is,
$y=f^{1}(x)\quad \Longleftrightarrow
\quad x=f(y)$
and
$x=f^{1}(y) \quad \Longleftrightarrow \quad y=f(x)$ 
A few facts:
 $f(f^{1}(\Box))=\Box$, no matter what you put into the box.
 The domain of $f^{1}$ is the range of $f$; and the range of
$f^{1}$ is the domain of $f$.
So, the independent variable of $f$ is now the dependent
variable of $f^{1}$ and vice versa. Graphically, this
means that the horizontal axis becomes vertical, and the
vertical axis horizontal.
 The function $f^{1}$ exists only if $f$ is onetoone.
Graphically, $f^{1}$ is a function if its graph passes
the vertical line test, which means that the graph of $f$ must
pass the horizontal line test.
If $f$ is not onetoone, we must restrict the domain of $f$
first.
For instance, if $f(x)=x+1$, then the inverse function is
$f^{1}(x)=x1$, since subtracting 1 undoes the effect of adding
1. If $f(x)=2x$, then $f^{1}(x)=x/2$, since dividing by 2
undoes the effect of multiplying by 2. We'll explore how to
find the formula for $f^{1}(x)$ in general on the next slide.
