*Section 4-5:
Inverse
Functions*
* *
*An inverse is the
operation
that takes you back to where you started. The inverse of
multiplication
is division, adding and subtracting, square and square root, etc.*
*For functions,
there
are two conditions for a function to be the inverse function:*
*
1) g(f(x)) = x for all x in the domain of f*
* *
*
2) f(g(x)) = x for all x in the domain of g*
* *
*Notice in both
cases
you will get back to the element that you started with, namely, x.*

*The notation used
to
indicate an inverse function is: f *^{-1}(x)
pronounced "f inverse". This
notation does not mean 1/f(x).

__Example__
* *
*1) If f(x) =
3x
- 1 and g(x) = (x + 1)/3, show that f and g are inverses to each other.*
*
To show that they are inverses, we must prove both of the above parts.*
*
g(f(x)) = g(3x - 1) = (3x - 1 + 1)/3 = 3x/3 = x*
*
f(g(x)) = f((x + 1)/3) = 3[(x + 1)/3] - 1 = x + 1 - 1 = x*
*Since both parts
work,
they are indeed inverses of each other.*

__To find a rule
for
the inverse function__
* *
*Find the inverse
function
for y = 5x + 2*
*To find the
inverse,
interchange x and y.*
*x = 5y + 2*
*Now isolate for y!!*
*x - 2 = 5y*
*(x - 2)/5 = y*
*We now have the
inverse!!*
*Notice, that this
inverse
make sense. The original problem had adding by two and the
inverse
is subtracting two. The original function had multiplying by five
and the inverse has division by five.*

__Graphs of
inverse
functions__
* *
*We have to make
sure
that the inverse is indeed a function. Not all functions will
have
inverses that are also functions. In order for a function to have
an inverse, it must pass the horizontal line test!!*
*Horizontal line test*
*If the graph of a
function
y = f(x) is such that no horizontal line intersects the graph in more
than
one point, then f has an inverse function.*
*This will make
sense
when we discover how to graph the inverse function. To graph the
inverse function, it is simply the reflection about the line y =
x.
Makes sense, because in order to get the graph, we interchange x and
y.
Recall from previous sections what the reflection about the line y = x
looks like. Any two points on the same horizontal line when
reflected
will be on the same vertical line. Can't have this because it
wouldn't
be a function. That's why the horizontal line test works.*

__Example__
* *
*1) Find the
equation
of f *^{-1}
and
graph f, f ^{-1},
and
y = x for f(x) = 2x - 5.
*
First, f(x) is a line and it passes the horizontal line test.*

*
Find the inverse: y = 2x - 5*

*
x = 2y - 5*

*
x + 5 = 2y*

*
(x + 5)/2 = y*

*2) Let f(x) = 9 - x*^{2}
for x __>__ 0

*
Find the equation for f *^{-1}(x)

*
Sketch the graph of f, f *^{-1},
and y = x.

* *

*
Notice that the equation is half of a parabola. Only the side to
the right of zero. If we tried to use the entire parabola, it
wouldn't
pass the horizontal line test.*

*
To find the equation: y = 9 - x*^{2}

*
x = 9 - y*^{2}

*
x - 9 = -y*^{2}
* 9 - x
=
y*^{2}
^{ ,
x < 9}

On to the last
section:
Back up and regroup: