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:
"f(x)" as "y":
I solve for "x
I get the
x-stuff on one side:
the trick: I factor out the x!
"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:
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) = x2 – 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.