Chapter 13 - Inverse Functions
### Chapter 13 - Inverse Functions

In the second part of this book on
Calculus, we shall be devoting our study to another type of function, the
exponential function and its close relative the Sine function. Before we
immerse ourselves in this complex and analytical study, we first need to
understand something about inverse functions.

The Inverse function is by
definition a function whose output becomes the input or the dependent
variable
becomes an independent variable. For example given the function:

Which is
Newton.s second law, or the force acting on a body of mass, m, is a
function of
the acceleration given to it. We are free to input any a and what we get
out is
the force. The inverse of this Force function, according to the
definition,
will give us the acceleration as a function of Force. This is done by
simply
solving for the independent variable, a:

Now I can let F
be anything and then find the acceleration as a function of it.

The inverse of a function, f(x), is
commonly written as,
. Now we will look at the more general case of graphing a
function and its inverse in the same co-ordinate planes. Given the
function
, to calculate its inverse we only have to solve this for x
to get
. Notice that we have not really changed the function at all,
we have only solved for the independent variable. The graph of these two
functions would be exactly the same. Our definition of the inverse
function
therefore has to be slightly modified. After finding the inverse of a
function
we just interchange x and y to get:

What does this do to the inverse function?
This essential flips the graph of f(x) about the line y=x such that for
every
point (x,y) there is a corresponding point (y,x) on the graph of the
inverse
function. Now both the functions can be graphed in the same x-y plane.

Remember that if we just solve for
the dependent, we are not changing the equation but merely re-writing it.
For
this reason its graph is the same. By flipping the x and y, we get another
function of x, whose relation to f(x) is that it has been graphed as
though the
x-axis were the y-axis and vice-versa. It is best we look at the two
graphs:

Notice how every
point (x, y) has a corresponding point (y,x) on the inverse function. The
graph
of the inverse function is therefore exactly the same as the original
function
except that the x and y-axis have been switched:

Since every point (x, y) has a
corresponding point (y,x) then any point y from the inverse function when
inputted in the original function should yield x:

Remember the a
function and its inverse are both function.s of x. The way they are
related is
that the inverse function represents the original function by just having
its
dependent and independent variable switched around. As you can see from
the
first graph, when the two function.s are graphed together, the inverse
function
contains all the point (x, y), of the first function, plotted as (y, x)
with
the exception that y is given as function of x. For this reason

What is
important to understand about the inverse function is that it is obtained
by
solving for the independent variable, then replacing it with y, to create
a
function that is also a function of x and can be graphed along with the
original function.

Now that we know how a function and
its inverse function are closely related, it brings us to the question,
how are
the derivatives related? Logic would tell us that instead of
we should just find
by taking the
reciprocal of the derivative. For example if we had:

The derivative
of the inverse function might be:

Or the
derivative of
is 1/2x. But this is not the case, the derivative is:

Let us examine
the graph of f(x) and its inverse function to see what exactly is going
on.

Note that at
x=2, the slopes are not reciprocals but are reciprocals only at y values
of on
the inverse function or through (x, f(x)) and (f^{-1}(f(x)), x).
Or the
point (3,9) will have a reciprocal slope at (9,3) since at this point x
and y
are reversed hence the slope becomes the reciprocal or
This is the
important point to understand about the function and its inverse, they
only
behave as opposites at point (a,b) and (b,a). This means that at point a
something different is going on. The question is then how can we find the
derivative of the inverse function with respect to the x-axis? Looking
again
at:

By replacing x
with y and y with x in this last expression we get:

What we have
just done is calculated the derivative of the inverse function only by
looking
at the original function and its derivative. The reason the derivative was
not
just the reciprocal of y=2x was because we forgot to do the following two
steps:

1) Replace x with its equivalent
expression in terms of y.

The slope in the
following graph is
at x=2 the slope
is (2)(2)=4

By replacing x
with
we can find the
derivative with respect to the** same
x-axis** but instead with a **y-value.**

At y=4 the slope
is
which is the answer
we got using x=2 instead.

Since the inverse function is
graphed in the same xy plane as
, we can find the derivative of the inverse function with
respect to the axis by taking the reciprocal of the expression
and then replacing
every y with x and vice-versa.

This last
expression is the derivative for the function.s inverse with respect to
the
x-axis.

To summarize we can state the
following theorem:

To find the
derivative of the inverse function,

1) Remember,
an inverse function is related to the main function in that if you reflect it over the line y=x, you will land on the main function.

2) First find
the derivative of f(x)

3) Replace any x
in the derivative with its y-equivalent, so as to be able to find the
derivative with any given y-value.

4) Take the
reciprocal of the derivative to get
so as to be able to
find the derivative with respect to the y-axis.

5) Since the
inverse function is graphed with respect to x, replace every y with x and
x
with y to find the derivative of the inverse function.

To summarize further:

For example
given
find the derivative of its inverse,