1

Functions and Models

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1.5

Inverse Functions and Logarithms

Copyright © Cengage Learning. All rights reserved.

Inverse Functions

3

Inverse Functions (1 of 14)

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.

Table 1

N as a function of t

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.

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Inverse Functions (2 of 14)

This function is called the inverse function of f, denoted by

Thus

is the time required for the population level to reach N.

The values of

can be found

by reading Table 1 from right to left or by consulting Table 2.

Table 2

t as a function of N

For instance,

because f (6) = 550.

Not all functions possess inverses.

5

Inverse Functions (3 of 14)

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 (x₁) ≠ f (x₂) whenever x₁ ≠ x₂

Figure 1

f is one-to-one; g is not.

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Inverse Functions (4 of 14)

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

1 Definition A function f is called a one-to-one function if it never takes on�the same value twice; that is.

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Inverse Functions (5 of 14)

If a horizontal line intersects the graph of f in more than one point, then we see from Figure 2 that there are numbers x₁ and x₂ such that f (x₁) = f (x₂).

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

Figure 2

This function is not one-to-one because f (x₁) = f (x₂).

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

Horizontal Line Test A function is one-to-one if and only if no horizontal line�intersects its graph more than once.

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Example 1

Is the function

one-to-one?

Solution 1:

If x₁ ≠ x₂, then

(two different numbers can’t have the same cube).

Therefore, by Definition 1,

Solution 2:

From Figure 3 we see that no horizontal line

intersects the graph of

Therefore, by the Horizontal Line Test, f is one-to-one.

Figure 3

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Inverse Functions (6 of 14)

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

2 Definition Let f be a one-to-one function with domain A and range B.

Then its inverse function

has domain B and range A and is defined by

for any y in B.

This definition says that if f maps x into y, then

maps y back into x. (If f were

not one-to-one, then

would not be uniquely defined.)

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Inverse Functions (7 of 14)

The arrow diagram in Figure 5 indicates that

reverses the effect of f.

Figure 5

Note that

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Inverse Functions (8 of 14)

For example, the inverse function of

because if

then

Caution

Do not mistake the −1 in

for an exponent. Thus

The reciprocal

could, however, be written as

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Example 3

If is a one-to-one function and f (1) = 5, f (3) = 7, and f (8) = −10, find

Solution:

From the definition of

we have

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Example 3 – Solution

The diagram in Figure 6 makes it clear how

reverses the effect of f in this case.

Figure 6

The inverse function reverses inputs and outputs.

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Inverse Functions (9 of 14)

The letter x is traditionally used as the independent variable, so when we

concentrate on

rather than on f, we usually reverse the roles of x and y in

Definition 2 and write

3

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

4

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Inverse Functions (10 of 14)

The first cancellation equation says that if we start with x, apply f, and then apply

we arrive back at x, where we started (see the machine diagram in Figure 7).

Figure 7

Thus

undoes what f does.

The second equation says that f undoes what

does.

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Inverse Functions (11 of 14)

For example, if

and so the cancellation equations

become

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

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Inverse Functions (12 of 14)

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

If we want to call the independent variable x, we then interchange x and y and

arrive at the equation

5 How to Find the Inverse Function of a One-to-One Function f

STEP 1 Write y = f (x).

STEP 2 Solve this equation for x in terms of y (if possible).

STEP 3

To express

as a function of x, interchange x and y.

The resulting equation is

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Inverse Functions (13 of 14)

The principle of interchanging x and y to find the inverse function also gives us

the method for obtaining the graph of

from the graph of f.

Since f (a) = b if and only if

the point (a, b) is on the graph of f if and only

if the point (b, a) is on the graph of

But we get the point (b, a) from (a, b) by reflecting about the line y = x. (See Figure 8.)

Figure 8

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Inverse Functions (14 of 14)

Therefore, as illustrated by Figure 9:

The graph of

is obtained by reflecting the graph of f about the line y = x.

Figure 9

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Logarithmic Functions

21

Logarithmic Functions (1 of 5)

If b > 0 and b ≠ 1, the exponential function

is either increasing or

decreasing and so it is one-to-one by the Horizontal Line Test. It therefore has

an inverse function

which is called the logarithmic function with base b

and is denoted by log_b.

If we use the formulation of an inverse function given by 3,

then we have

6

22

Logarithmic Functions (2 of 5)

Thus, if x > 0, then log_b x is the exponent to which the base b must be raised to give x.

For example,

The cancellation equations (4), when applied to the functions

become

7

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Logarithmic Functions (3 of 5)

The logarithmic function log_b has domain (0,∞) and range

Its graph is the

reflection of the graph of

about the line y = x.

Figure 11 shows the case where b > 1. (The most important logarithmic functions have base b > 1.)

Figure 11

The fact that

is a very rapidly

increasing function for x > 0 is reflected in the fact that y = log_b x is a very slowly increasing function for x > 1.

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Logarithmic Functions (4 of 5)

Figure 12 shows the graphs of y = log_b x with various values of the base b > 1.

Since log_b 1 = 0, the graphs of all logarithmic functions pass through the point (1, 0).

Figure 12

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Logarithmic Functions (5 of 5)

The following properties of logarithmic functions follow from the corresponding properties of exponential functions.

Laws of Logarithms If x and y are positive numbers, then

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Example 6

Use the laws of logarithms to evaluate

Solution:

Using Law 2, we have

because

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Natural Logarithms

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Natural Logarithms (1 of 4)

Of all possible bases b for logarithms, we will see that the most convenient choice of a base is the number e.

The logarithm with base e is called the natural logarithm and has a special notation:

If we set b = e and replace log_e with “ln” in (6) and (7), then the defining properties of the natural logarithm function become

8

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Natural Logarithms (2 of 4)

9

In particular, if we set x = 1, we get

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Natural Logarithms (3 of 4)

Combining Property 9 with Law 3 allows us to write

Thus a power of x can be expressed in an equivalent exponential form; we will find this useful in the chapters to come.

10

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Example 7

Find x if ln x = 5.

Solution 1:

From (8) we see that

Therefore

(If you have trouble working with the “ln” notation, just replace it by log_e. Then

the equation becomes log_e x = 5; so, by the definition of logarithm,

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Example 7 – Solution

Solution 2:

Start with the equation

ln x = 5

and apply the exponential function to both sides of the equation:

But the second cancellation equation in (9) says that

Therefore

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Natural Logarithms (4 of 4)

The following formula shows that logarithms with any base can be expressed in terms of the natural logarithm.

11 Change of Base Formula For any positive number b (b ≠ 1). we have

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Example 11

Evaluate log₈ 5 correct to six decimal places.

Solution:

Formula 11 gives

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Graph and Growth of the Natural Logarithm

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Graph and Growth of the Natural Logarithm (1 of 4)

The graphs of the exponential function

and its inverse function, the natural

logarithm function, are shown in Figure 13.

Figure 13

The graph of y = ln x is the reflection of the graph of

37

Graph and Growth of the Natural Logarithm (2 of 4)

In common with all other logarithmic functions with base greater than 1, the natural logarithm is an increasing function defined on (0, ∞) and the y-axis is a vertical asymptote. (This means that the values of ln x become very large negative as x approaches 0.)

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Example 12

Sketch the graph of the function y = ln(x − 2) − 1.

Solution:

We start with the graph of y = ln x as given in Figure 13.

We shift it 2 units to the right to get the graph of y = ln(x − 2) and then we shift it 1 unit downward to get the graph of y = ln(x − 2) − 1. (See Figure 14.)

Figure 14

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Graph and Growth of the Natural Logarithm (3 of 4)

Although ln x is an increasing function, it grows very slowly when x > 1. In fact, ln x grows more slowly than any positive power of x.

​

To illustrate this fact, we graph

in Figure 15 and 16.

Figure 15

Figure 16

40

Graph and Growth of the Natural Logarithm (4 of 4)

You can see that the graphs initially grow at comparable rates, but eventually the root function far surpasses the logarithm.

41

Inverse Trigonometric Functions

42

Inverse Trigonometric Functions (1 of 10)

When we try to find the inverse trigonometric functions, we have a slight difficulty: Because the trigonometric functions are not one-to-one, they don’t have inverse functions.

The difficulty is overcome by restricting the domains of these functions so that they become one-to-one.

43

Inverse Trigonometric Functions (2 of 10)

You can see from Figure 17 that the sine function y = sin x is not one-to-one (use the Horizontal Line Test).

Figure 17

However, if we restrict the domain to the interval

then the function is one-to-one and all values in the range of y = sin x are attained (see Figure 18).

Figure 18

44

Inverse Trigonometric Functions (3 of 10)

The inverse function of this restricted sine function f exists and is denoted by

or arcsin. It is called the inverse sine function or the arcsine function.

Since the definition of an inverse function says that

we have

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Inverse Trigonometric Functions (4 of 10)

So, if −1 ≤ x ≤ 1,

is the number between

whose sine is x.

46

Example 13

Evaluate (a)

and (b)

Solution:

(a) We have

because

lies between

47

Example 13 – Solution

(b) Let

Then we can draw a right triangle with angle θ as in Figure 19 and deduce from

the Pythagorean Theorem that the third side has length

This enables us to read from the triangle that

Figure 19

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Inverse Trigonometric Functions (5 of 10)

The cancellation equations for inverse functions become, in this case,

49

Inverse Trigonometric Functions (6 of 10)

The inverse sine function,

has domain [−1, 1] and range

and its graph, shown in Figure 20, is obtained from that of the restricted sine function (Figure 18) by reflection about the line y = x.

Figure 18

Figure 20

50

Inverse Trigonometric Functions (7 of 10)

The inverse cosine function is handled similarly. The restricted cosine function f (x) = cos x, 0 ≤ x ≤ π , is one-to-one (see Figure 21) and so it has an

inverse function denoted by

Figure 21

y = cos x, 0 ≤ x ≤ π

51

Inverse Trigonometric Functions (8 of 10)

The cancellation equations are

The inverse cosine function,

has domain [−1, 1] and range [0, π]. Its graph is shown in Figure 22.

Figure 22

52

Inverse Trigonometric Functions (9 of 10)

The tangent function can be made one-to-one by restricting it to the interval

Thus the inverse tangent function is defined as the inverse of

the function

(See Figure 23.) It is denoted by

or arctan.

Figure 23

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Inverse Trigonometric Functions (10 of 10)

The remaining inverse trigonometric functions are summarized here.

12

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