2.4  INVERSE  FUNCTIONS

     Two real functions f and g are called inverse functions if the two equations

y=f(x),   x=g(y)

Have the same graphs in the (x, y) plane. That is, a point (x, y) is on the curve y=f(x) if, and only if, it is on the curve x=g(y). (in general, the graph of the equation x=g(y) is different from the graph of y=g(x), but is the same as the graph of y=f(x); see Figure 2.4.1.)

Figure 2.4.1 inverse functions

For example, the function y=x², x ≥ 0, has the inverse function x=_______; the function y=x ³ has the inverse function x=______.

If we think of f as a black box operating on an input x to produce an output f(x), the inverse function g is a black box operating on the output f(x) to undo the work of f and produce the original input x  (see Figure 2.4.2).

Figure 2.4.2

Many functions, such as y=x2, do not have inverse functions. In Figure 2.4.3, we see that x is not a function of y because at y=1, x has the two values x=1 and x= -1.

Often one can tell whether a function f has an inverse by looking at its graph. If there is a horizontal line y=c which cuts the graph at more than one point, the function f has no inverse. (see figure 2.4.3.) if no horizontal line cuts the graph at more than one point, then f has an inverse function g. Using this rule, we can see in figure 2.4.4 that the functions y=| x | and y = _______________ do not have inverses.

Figure 2.4.3

Figure  2.4.4                   No inverse functions

Table 2.4.1 shows some familiar functions which do have inverses.

Note that in each case,

Suppose the (x, y) plane is flipped over about the diagonal line y=x. This will make the x-and y-axes change places, forming the (y,x) plane. If f has an inverse function g, the graph of the function y=f(x) will become the graph of the inverse function x=g(y) in the (y, x) plane, as shown in Figure 2.4.5.

The following rule shows that the derivatives of inverse functions are always reciprocals of each other.

INVERSE  FUNCTION  RULE

   Suppose f and g are inverse functions, so that the two equations

                    y=f(x)     and    x=g(y)

   have the same graphs. If both derivatives f '(x) and g '(y) exist and are nonzero,

then                     

that is,    

PROOF  Let Δx be a nonzero infinitesimal and let Δy be the corresponding change in y. Then Δy is also infinitesimal because f '(x) exists and is nonzero because f(x) has an inverse function. By the rules for standard parts,

Therefore  

(a)

(b)

(c)

(d)

Figure 2.4.5

The formula

in the Inverse Function Rule is not as trivial as it looks. A more complete statement is

____  computed with x the independent variable

= ________ computed with y the independent variabe.

Sometimes it is easier to compute dx/dy than dy/dx, and in such cases the Inverse Function Rule is a useful method.

EXAMPLE 1  Find dy/dx where x=1+y -3.

          Before solving the problem we note that

     So x and y are inverse functions of each other. We want to find

With x the independent variable. This looks hard, but it is easy to compute

With y the independent variable.

SOLUTION   ____ = -3y -4,

We can write dy/dx in terms of x by substituting,

EXAMPLE  2  Find dy/dx where x= y5 + y3 + y. Compute dy/dx at the point(3,1).

       Although we cannot solve the equation explicitly for y as a function of x, we can see from the graph in Figure 2.4.6 that there is an inverse function y=f(x).

Figure 2.4.6

By the Inverse Function Rule,

This time we must leave the answer in term of y. At the point (3,1), we substitute 1 for y and get dy/dx = 1/9.

For y≥ 0, the function x= y n has the inverse function y= x1/n. In the next theorem, we use the Inverse Function Rule to find a new derivative, that of y= x1/n.

THEOREM  1

     If n is a positive integer and

y= x1/n.

     then                   

Remember that y= x1/n. is defined for all x if n is odd and for  x > 0 if n is even.

The derivative______ is defined for x≠ 0 if n is odd and for x > 0 if n is even.

If we are willing to assume that dy/ dx exists, then we can quickly find dy/dx by the Inverse Function Rule.

Here is a longer but complete proof which shows that dy/dx exists and computes its value.

PROOF  OF  THEOREM 1

              Let x≠ 0 and let Δx be nonzero infinitesimal. We first show that

Δy= (x + Δx) 1/n - x1/n

           is a nonzero infinitesimal. Δy ≠ 0 because x+Δ x ≠ x. The standard part of

Δy is

st( Δy) = st(( x+ Δ x)1/n) - st(x1/n )

                                    =x1/n - x1/n = 0.

Therefore y is nonzero infinitesimal.

Now                  

Therefore

Figure 2.4.7

Figure 2.4.7 shows the graphs of y=x1/3 and y=x1/4. At x =0, the curves are vertical and have no slope.

EXAMPLE  3  Find the derivatives of y=x1/n for n=2, 3,4.

Using Theorem 1 we can show that the Power Rule holds when the exponent is any rational number.

POWER  RULE  FOR  RATIONAL  EXPONENTS

Let y=xr where r is a rational number. Then whenever x>0,

PROOF  Let r=m/n where m and n are integers, n>0. Let

u=x 1/n,  y=um.

Then                __________________

and

EXAMPLE  4  Find dy/dx where

EXAMPLE  5  Find dy/dx where

                        ______________________.

Let                 u=2+x3/2,   y= u -1.

Then          

PROBLEMS  FOR  SECTION 2.4

In Problems 1-16, find dy/dx.

1       x=3y3 + 2y                      2   x = y2 + 1,    y>0

3       x= 1 - 2y2 , y>0                  4   x=2y5 + y3 + 4

5       x= (y2 +2)-1 , y>0                 6   _______

7       y=x 4/3                                   8   _______

9       ____________                   10  y= (2x 1/3+ 1) 3 

11      y= 1+ 2x 1/3 + 4x 2/3 +6x  12  y=x -1/4 + 3x -3/4

13      y= (x 5/3 - x) -2                           14   x= y+_____

15     x= 3x 1/3 +2y,  y > 0               16      x= 1/(1 + ____)

In Problems 17-25, find the inverse function y and its derivative dy/dx as functions of x.

17    x=ky+c,  k≠ 0                     18    x= y3 + 1

19    x=2y2+1, y≥ 0                      20    x=2y2+1, y≤0   

21    x=y4 - 3, y≥ 0                       22    x=y2+3y-1,  y ≥ ____

23    x=y4+y2 +1 , y≥ 0                    24   x = 1/y2 +1/y -1 ,  y > 0  

25    _______________

□26  show that no second degree polynomial x= ay2 + by +c has an inverse function.

□27  show that x=ay2 + by + c, y ≥ -b/2a, has an inverse function. What does its graph look

      like?

□28  Prove that a function y=f(x) has an inverse function if and only if whenever x1≠ x2,

f(x1)__ f(x2)