## Inverse Functions

Table 1 gives data from an experiment in which a bacteria culture started with 100 bacteria in a limited nutrient

medium; the size of the bacteria population was recorded at hourly intervals.

*The number of bacteria N is*

*a function of the time t: N = f(t).*

Suppose, however, that the
biologist changes her point of
view and becomes interested
in the time required for the
population to reach various
levels. In other words, she is
*thinking of t as a function of N.*

2

## Inverse Functions

*This function is called the inverse function of f, denoted*
*by f*^{–1}*, and read “f inverse.” Thus t = f*^{–1}*(N) is the time *
*required for the population level to reach N.*

*The values of f*^{–1 }can be found
by reading Table 1 from right
to left or by consulting Table 2.

*For instance, f*^{–1}(550) = 6
*because f(6) = 550.*

Not all functions possess

inverses. *t as a function of N*

**Table 2**

## Inverse Functions

*Let’s compare the functions f and g whose arrow diagrams *
are shown in Figure 1.

*Note that f never takes on*

the same value twice (any two

*inputs in A have different outputs),*
*whereas g does take on the same*
value twice (both 2 and 3 have

the same output, 4).

In symbols,

*g(2) = g(3)*

*but f(x* *) ≠ f(x* ) *whenever x* *≠ x* *f is one-to-one; g is not.*

4

## Inverse Functions

*Functions that share this property with f are called *
*one-to-one functions.*

## Inverse Functions

If a horizontal line intersects the
*graph of f in more than one point,*
then we see from Figure 2 that
*there are numbers x*_{1} *and x*_{2}
*such that f(x*_{1}*) = f(x*_{2}).

*This means that f is not one-to-one.*

Therefore we have the following geometric method for determining whether a function is one-to-one.

This function is not one-to-one
*because f(x*_{1}*) = f(x*_{2}).

**Figure 2**

6

## Example 1

*Is the function f(x) = x*^{3} one-to-one?

Solution 1:

*If x*_{1} *≠ x*2*, then x*_{1}^{3} *≠ x*23 (two different numbers can’t have
the same cube).

*Therefore, by Definition 1, f(x) = x*^{3} is one-to-one.

Solution 2:

From Figure 3 we see that no horizontal line intersects the

*graph of f(x) = x*^{3} more than once.

Therefore, by the Horizontal

*Line Test, f is one-to-one.* ^{f(x) = x}^{3}is one-to-one.

**Figure 3**

## Inverse Functions

One-to-one functions are important because they are precisely the functions that possess inverse functions according to the following definition.

*This definition says that if f maps x into y, then f*^{–1} *maps y *
*back into x. (If f were not one-to-one, then f*^{–1} would not be
uniquely defined.)

8

## Inverse Functions

*The arrow diagram in Figure 5 indicates that f*^{–1} reverses
*the effect of f.*

Note that

**Figure 5**

## Inverse Functions

*For example, the inverse function of f(x) = x*^{3} *is f*^{–1}*(x) = x*^{1/3}
*because if y = x*^{3}, then

*f*^{–1}*(y) = f*^{–1}*(x*^{3}*) = (x*^{3})^{1/3 }*= x*
**Caution**

*Do not mistake the –1 in f*^{–1} for an exponent. Thus
*f*^{–1}*(x) does not mean*

*The reciprocal 1/f(x) could, however, be written as [f(x)]*^{–1}.

10

## Example 3

*If f(1) = 5, f(3) = 7, and f(8) = –10, find f*^{–1}*(7), f*^{–1}(5), and
*f*^{–1}(–10).

Solution:

*From the definition of f*^{–1 }we have

*f*^{–1}(7) = 3 because *f(3) = 7*
*f*^{–1}(5) = 1 because *f(1) = 5*
*f*^{–1}(–10) = 8 because *f(8) = –10*

*Example 3 – Solution*

*The diagram in Figure 6 makes it clear how f*^{–1 }reverses the
*effect of f in this case.*

cont’d

The inverse function reverses inputs and outputs.

**Figure 6**

12

## Inverse Functions

*The letter x is traditionally used as the independent variable, *
*so when we concentrate on f*^{–1 }*rather than on f, we usually *
*reverse the roles of x and y in Definition 2 and write*

*By substituting for y in Definition 2 and substituting for *
**x in (3), we get the following cancellation equations:**

## Inverse Functions

*The first cancellation equation says that if we start with x, *
*apply f, and then apply f*^{–1}*, we arrive back at x, where we *
started (see the machine diagram in Figure 7).

*Thus f*^{–1} *undoes what f does.*

*The second equation says that f undoes what f*^{–1} does.

**Figure 7**

14

## Inverse Functions

*For example, if f(x) = x*^{3}*, then f*^{–1}*(x) = x*^{1/3} and so the
cancellation equations become

*f*^{–1}*(f(x)) = (x*^{3})^{1/3} *= x*
*f(f*^{–1}*(x)) = (x*^{1/3})^{3} *= x*

These equations simply say that the cube function and the cube root function cancel each other when applied in

succession.

## Inverse Functions

Now let’s see how to compute inverse functions.

*If we have a function y = f(x) and are able to solve this *
*equation for x in terms of y, then according to Definition 2 *
*we must have x = f*^{–1}*(y).*

*If we want to call the independent variable x, we then *
*interchange x and y and arrive at the equation y = f*^{–1}*(x).*

16

## Inverse Functions

*The principle of interchanging x and y to find the inverse *

function also gives us the method for obtaining the graph of
*f*^{–1} *from the graph of f.*

*Since f(a) = b if and only if f*^{–1}*(b) = a, the point (a, b) is on *
*the graph of f if and only if the point (b, a) is on the graph*
*of f*^{–1}.

*But we get the point (b, a)*

*from (a, b) by reflecting about*
*the line y = x. (See Figure 8.)*

**Figure 8**

## Inverse Functions

Therefore, as illustrated by Figure 9:

18

## The Calculus of Inverse Functions

## The Calculus of Inverse Functions

Now let’s look at inverse functions from the point of view of calculus.

*Suppose that f is both one-to-one and continuous. We think *
of a continuous function as one whose graph has no break
in it. (It consists of just one piece.)

*Since the graph of f*^{–1 }is obtained
*from the graph of f by reflecting *
*about the line y = x, the graph of *
*f* ^{–1 }has no break in it either (see
Figure 9).

20

## The Calculus of Inverse Functions

*Thus we might expect that f*^{–1 }is also a continuous function.

This geometrical argument does not prove the following theorem but at least it makes the theorem plausible.

*Now suppose that f is a one-to-one differentiable function. *

Geometrically we can think of a differentiable function as one whose graph has no corner or kink in it.

## The Calculus of Inverse Functions

*We get the graph of f*^{–1} *by reflecting the graph of f about *
*the line y = x, so the graph of f*^{–1} has no corner or kink in it
either.

*We therefore expect that f*^{–1} is
also differentiable (except

where its tangents are vertical).

In fact, we can predict the

*value of the derivative of f*^{–1 }at
a given point by a geometric
argument.

*In Figure 11 the graphs of f and its * _{Figure 11}

22

## The Calculus of Inverse Functions

*If f(b) = a, then f*^{–1}*(a) = b and (f*^{–1})′(a) is the slope of the
*tangent line L to the graph of f*^{–1} *at (a, b), which is*

*∆y/∆x. *

*Reflecting in the line y = x has the effect of interchanging *
*the x- and y-coordinates. So the slope of the reflected line *

*[the tangent to the graph of f at (b, a)] is ∆x/∆y.*

*Thus the slope of L is the reciprocal of the slope of , that *
is,

## The Calculus of Inverse Functions

24

## Example 6

*Although the function y = x*^{2}, is not one-to-one and
therefore does not have an inverse function, we can turn it
into a one-to-one function by restricting its domain.

*For instance, the function f(x) = x*^{2}, 0 ≤ x ≤ 2, is one-to-one
(by the Horizontal Line Test) and has domain [0, 2] and

*range [0, 4]. (See Figure 12.) Thus f has an inverse *
*function f*^{–1} with domain [0, 4] and range [0, 2].

**Figure 12**

*(a) y = x*^{2}, *(b) f (x) = x*^{2}, 0 ≤ x ≤ 2

## Example 6

*Without computing a formula for (f*^{–1})′ we can still calculate
*(f*^{–1})′(1). Since f(1) = 1, we have f ^{–1}*(1) = 1. Also f*^{–1}*(x) = 2x. *

So by Theorem 7 we have

*In this case it is easy to find f*^{–1 }*explicitly. In fact, f*^{–1}*(x) = *

0 ≤ x ≤ 4. [In general, we could use the method given by (5).]

Then which agrees with the preceding computation.

cont’d

26

## Example 6

*The functions f and f*^{–1} are graphed in Figure 13.

cont’d

**Figure 13**