, at least when
. These two functions have the same graph if we plot g with its independent variable where the y axis normally goes, rather than plotting the input variable of g[y] on the horizontal scale.
The inverse of a function y=f[x] is the function x=g[y] whose "rule undoes what the rule for f does." For example, if y=f[x]=x2, then
, at least when
. These two functions have the same graph if we plot g with its independent variable where the y axis normally goes, rather than plotting the input variable of g[y] on the horizontal scale.
We can take the usual convention with input variable on the horizontal axis for both functions. This makes the graph of the inverse function the "mirror image" across y=x. For this project you will find it best to plot both functions with the x and y axes in the same place.
Example 21.1 A Partial Inverse by Algebra
Often we can find an inverse function or partial inverse function by solving equations.
For example, if y=x2+1, then x2=y-1 and
when
and
. The negative square root also satisfies the original equation,
.
The algebra of inverse functions can be tricky, but the calculus of their derivatives is much easier - just look in the infinitesimal microscope.
21.2
Example 21.2
The Derivative of the Inverse
It is sometimes easier to compute the derivative of the inverse function and invert for the derivative of the function itself.
For example, if y=x2+1 and
Example 21.3
Derivative of Log by Inverse Functions
The derivative of
Here is some practice at using the formula for the derivative of the inverse function.
Conversion of variables will be a little more trouble with trig functions, but how else would you find the derivative?
The Inverse Function Rule can be used numerically even when the formula for the inverse function is not known.
The next exercise shows you what we mean.
Show that
The important Inverse Function Theorem that says that if a function has a non-zero derivative, then at least over an interval, the curve y=f[x] has an inverse function with the same graph (when the same axes are used for both plots). The "proof" (as opposed to the rule that assumes you know g) uses the view in an infinitesimal microscope to compute an approximation to the inverse function.
You see now that computation of the derivative of the inverse function is obvious (at least in the dependent variable) once you know the derivative of y=f[x], but it may not be so obvious how to compute the inverse function itself: How would you compute arctangent or arcsine yourself? This project gives an easy method.
Example 21.4
A Nonelementary Inverse
Some functions do not have expressions for their inverses.
Let
What is the microscopic view of y=xx at (x,y)=(1,1)? After you have answered this question, use the computer to plot the function.
The slope table from zero to infinity is down, over up, so it is "clear" that once
What is
It is clear graphically from the previous exercise that y=f[x] has an inverse on either the interval (0,1/e) or
Example 21.5
Microscopic Approximation to the Inverse
Suppose we have a function y=f[x] and know that f'[x] exists on an interval around a point x=x0 and we know the values y0=f[x0] and m=f'[x0]. In a microscope we would see the graph (Figure 21.11)
The point (dx,dy)=(0,0) in local coordinates is at (x0,y0) in regular coordinates.
Suppose we are given y near y0,
What x value, x1, corresponds to dx=dy/m? The answer is dx=x1-x0 with x1 unknown.
Solve for the unknown,
The graph of y=f[x] appears to be the parallel line above the tangent, because we have only moved x a small amount and f'[x] is continuous by Theorem CD 5.3. We don't know how to compute x=g[y] necessarily, but we do know how to compute y1=f[x1]. Suppose we have computed this and focus our microscope at (x1,y1) seeing Figure 21.13.
We still have the original
What x value, x2, corresponds to dx=dy/m? The answer is dx=x2-x1 with x2 unknown.
Solve for the unknown,
We know that
Modify the program to compute arctangent, but without using the built-in function.
Take
You will see in the computer program that the approximation for the arcsine converges very rapidly for nearby values of
Experiment with the program InverseFctHelp to see how far away from 0.5 you can take y and still successfully compute arcsine.
Experiment with the program InverseFctHelp to see how far away from 0 you can take y and still successfully compute arctangent.
The general successive approximation scheme to compute the inverse function x=g[y] is
Prove that
In order for the
21.3
Conclude your project by computing the inverse function x=g[y] when y=f[x]=xx. Explain why your the computer program is a convergent approximation procedure (even though there is no elementary expression for g).
There is more information on the Inverse Function Theorem in the Mathematical Background book on CD.
at corresponding (x,y) pairs.
when
, then
. The inverse function rule says
We converted the expression
into an expression in the independent variable for g at the last step.
We can compute
directly from rules using the formula
to see that these rules agree.
Since
, the Power Rule and Chain Rule give
Notice that the first computation is somewhat easier but used conversion of variables at the last step.
follows from the Inverse Function Rule and the rule for the natural exponential.
We have only given these rules without proof, but we observe here that we only need to prove one of the two.
The inverse of
is y=ex and has derivative
, therefore
We know that if
, then
. (Hint: 
, so you can use the Chain Rule and Product Rule.
Also see text Section 5.3 on CD.) Use the Inverse Function Rule to compute
when
. We want you to express your answer in terms of the independent variable for the arctangent, y, so you need to use identities to convert
into a function of y. We know that
and
, so we can express
in terms of y.
.) How could Bolzano's Intermediate Value Theorem 20.2 in the Mean Value Math Police project (or the Math Background) for f tell you g exists? Why can't you find a formula for g?
by using numerics on the expression for g'[y] in terms of its dependent variable. (Note that f[1]=3.)
This may be written using
, so
, and f[x] has derivative
(x>1/e), the function is increasing and there is an inverse. (Also, there is another inverse from zero until x=1/e.)
? The answer shows why people sometimes write 00=1.
. It turns out that the inverse function x=g[y] can not be expressed in terms of any of the classical functions.
In other words, there is no formula for g[y]. (This is similar to the non-elementary integrals in the computer program SymbolicIntegr. We can compute them numerically with NIntegrate, but there is no elementary expression for them.
Computer algebra systems have a non-elementary function
that can be used to express the inverse.) We look in an infinitesimal microscope to approximate inverse functions in general.
Figure 21.1: Small View of y=f[x] at (x0,y0) 
. In the microscope, this appears on the dy axis at the local coordinate dy=y=y0. The corresponding dx value is easily computed by inverting the linear approximation
Does this value of x=x1 satisfy y=f[x1] for the value of y we started with? We wouldn't think it would be exact because we computed linearly and we only know the approximation
We know that the error
is small when
is small, so we would have to move the microscope to see the error.
Moving along the tangent line until we are centered over x1, we might see Figure 21.12.
Figure 21.2: Small View at (x1,y) 
Figure 21.3: Small View at (x1,y1) 
and thus y still appears on the new view at dy=y-y1. The corresponding dx value is easily computed by inverting the linear approximation
This is the same computation we did above to go from x0 to x1; in fact, this gives a discrete dynamical system of successive approximations
at
. We also know that
. Compute a few iterates of the approximation procedure above for
, 
and check your work with the computer program InverseFctHelp on our website.
, (x0,y0)=(0,0). Use the built-in arctangent to check your work.
This is how you could compute arctangent yourself.
.
What is a value of y not near y=0.5 for which you do not expect the approximation procedure for arcsine to converge? (Hint: When is
?)
We want to show that this tends toward the right value
for the inverse function g corresponding to f if and only if
is an equilibrium point of the dynamical system xn+1=G[xn] above.
, we need to have a stable dynamical system.
If you know that there is an equilibrium point
, apply the Nonlinear Stability Theorem of text Chapter CD 20 to show that it is an attractor.
What do you need to know about
compared with f'[x0] for stability?