Find the inverse
f(x)
= ^{(x – 2)} / _{(x + 2)}, where
x
does not equal –2. Is
the inverse a function?

First, I recognize
that f(x)
is a rational function.
Here's its graph:

The restriction on the
domain comes from the fact that I can't divide by zero, so x
can't be equal to
–2.
I usually wouldn't bother writing down the restriction, but it's helpful
here because I need to know the domain and range of the inverse. Note
from the picture (and recalling the concept of horizontal
asymptotes) that
y
will never equal
1.
Then the domain is "x
is not equal to
–2" and the
range is " y
is not equal to
1". For the
inverse, they'll be swapped: the domain will be "x
is not equal to
1" and the range
will be "y
is not equal to
–2". Here's
the algebra:

The
original function:

I
rename "f(x)"
as "y":

Then
I solve for "x
=":

I
get the x-stuff
on one side:

Here's
the trick: I factor out the x!

Then
I switch x
and y:

And
rename "y"
as "f-inverse";
the domain restriction comes from the fact that this is a
rational function.

Since the inverse
is just a rational function, then the inverse is indeed a function.

Here's the graph:

Then the
inverse is y
= ^{(–2x – 2)} / _{(x
– 1)},
and the inverse is also a function, with domain of all xnot equal
to
1and
range of all ynot equal
to
–2.

Find the inverse
of f(x)
= x^{2} – 3x + 2, x< 1.5

With the domain
restriction, the graph looks like this:

From what I know
about graphing
quadratics,
the vertex is at (x,
y) = (1.5, –0.25),
so this graph is the left-hand "half" of the parabola.