1. 1.6 Inverse Functions and Logarithms
FUNCTIONS AND MODELS
1.6
Inverse Functions
and Logarithms
REVISIT OBJECTIVE.
In this section, we will learn about:
Inverse functions and logarithms.

2. INVERSE FUNCTIONS
The table 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).

3. In other words, she is thinking of t as a function of N.
INVERSE FUNCTIONS
However, suppose 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.
This function is called the inverse function of f.
It is denoted by f -1 and read “f inverse.”

4. INVERSE FUNCTIONS
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 the first table from right to left or by consulting the second table.
For instance, f -1(550) = 6, because f(6) = 550.

5. Not all functions possess inverses.
INVERSE FUNCTIONS
Not all functions possess inverses.
Let’s compare the functions f and g whose arrow diagrams are shown.

6. Note that f never takes on the same value twice.
INVERSE FUNCTIONS
Note that f never takes on the same value twice.
Any two inputs in A have different outputs.
However, g does take on the same value twice.

7. In symbols, g(2) = g(3) but f(x1) ≠ f(x2) whenever x1 ≠ x2
INVERSE FUNCTIONS
In symbols, g(2) = g(3) but f(x1) ≠ f(x2) whenever x1 ≠ x2
Functions that share this property with f are called one-to-one functions.

8. That is, f(x1) ≠ f(x2) whenever x1 ≠ x2
ONE-TO-ONE FUNCTIONS
Definition 1
A function f is called a one-to-one function if it never takes on the same value twice.
That is, f(x1) ≠ f(x2) whenever x1 ≠ x2

9. ONE-TO-ONE FUNCTIONS
If a horizontal line intersects the graph of f in more than one point, then we see from the figure that there are numbers x1and x2 such that f(x1) = f(x2).
This means f is not one-to-one.
HORIZONTAL LINE TEST

10. Is the function f(x) = x3 one-to-one?
ONE-TO-ONE FUNCTIONS
Example 1
Is the function f(x) = x3 one-to-one?
If x1 ≠ x2, then x13 ≠ x23.
Two different numbers can’t have the same cube.
So, by Definition 1, f(x) = x3 is one-to-one.
From the figure, we see that no horizontal line intersects the graph of f(x) = x3 more than once.

11. ONE-TO-ONE FUNCTIONS
Example 2
Is the function g(x) = x2 one-to-one?
The function is not one-to-one.
This is because, for instance, g(1) = 1 = g(-1) and so 1 and -1 have the same output.
From the figure, we see that no horizontal line intersects the graph of f(x) = x3 more than once.

12. ONE-TO-ONE FUNCTIONS
Definition 2
Let f be a one-to-one function with domain A and range B.
Then, its inverse function f -1 has domain B and range A and is defined by
for any y in B.
Do not mistake the -1 in f -1 for an exponent. Thus, f -1(x) does not mean

13. ONE-TO-ONE FUNCTIONS
Example 3
This diagram makes it clear how f -1 reverses the effect of f in this case.

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 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).

15. Now, let’s see how to find the inverse
INVERSE FUNCTIONS
Definition 5
Now, let’s see how to find the inverse
function of a one-to-one function f.
Write y = f(x).
Solve this equation for x in terms of y (if possible).
To express f -1 as a function of x, interchange x and y.
The resulting equation is y = f -1(x).

16. Find the inverse function of f(x) = x3 + 2.
INVERSE FUNCTIONS
Example 4
Find the inverse function of
f(x) = x3 + 2.
By Definition 5, we first write: y = x3 + 2.
Then, we solve this equation for x :
Finally, we interchange x and y :
So, the inverse function is:

17. Thus, the graph of f -1 is obtained by
INVERSE FUNCTIONS
Thus, the graph of f -1 is obtained by
reflecting the graph of f about the line
y = x.

18. and its inverse function using the same coordinate axes.
INVERSE FUNCTIONS
Example 5
Sketch the graphs of
and its inverse function using the same
coordinate axes.

19. First, we sketch the curve (the top half of the parabola y2 = -1 -x,
INVERSE FUNCTIONS
Example 5
First, we sketch the curve
(the top half of the parabola y2 = -1 -x,
or x = -y2 - 1).
Then, we reflect
about the line y = x
to get the graph of f -1.

20. If a > 0 and a ≠ 1, the exponential function
LOGARITHMIC FUNCTIONS
If a > 0 and a ≠ 1, the exponential function
f(x) = ax is either increasing or decreasing,
so it is one-to-one by the Horizontal Line Test.
Thus, it has an inverse function f -1, which
is called the logarithmic function with base a
and is denoted by loga.

21. Thus, if x > 0, then logax is the exponent
LOGARITHMIC FUNCTIONS
Thus, if x > 0, then logax is the exponent
to which the base a must be raised
to give x.
For example, log = - 3 because = 0.001

22. The logarithmic function loga has domain and range .
LOGARITHMIC FUNCTIONS
The logarithmic function loga has
domain and range
Its graph is the reflection of the graph of y = ax about the line y = x.

23. The figure shows the graphs of y = logax
LOGARITHMIC FUNCTIONS
The figure shows the graphs of y = logax
with various values of the base a > 1.
Since loga1 = 0, the graphs of all logarithmic functions pass through the point (1, 0).

24. If x and y are positive numbers, then 1. 2. 3.
LAWS OF LOGARITHMS
If x and y are positive numbers, then 1. 2. 3.

25. Use the laws of logarithms to evaluate log280 - log25.
Example 6
Use the laws of logarithms to evaluate
log280 - log25.
Using Law 2, we have
because 24 = 16.

26. The logarithm with base e is called
NATURAL LOGARITHM
The logarithm with base e is called
the natural logarithm and has a special
notation:

27. Solve the equation e5 - 3x = 10.
NATURAL LOGARITHMS
Example 8
Solve the equation e5 - 3x = 10.
We take natural logarithms of both sides of the equation and use Definition 9:
As the natural logarithm is found on scientific calculators, we can approximate the solution— to four decimal places: x ≈

28. Express as a single logarithm. NATURAL LOGARITHMS Example 9
Using Laws 3 and 1 of logarithms, we have:

29. For any positive number a (a ≠ 1), we have:
CHANGE OF BASE FORMULA
Formula 10
For any positive number a (a ≠ 1),
we have:

30. Evaluate log8 5 correct to six decimal places.
NATURAL LOGARITHMS
Example 10
Evaluate log8 5 correct to six
decimal places.
Formula 10 gives:

31. Sketch the graph of the function y = ln(x - 2) -1.
NATURAL LOGARITHMS
Example 11
Sketch the graph of the function
y = ln(x - 2) -1.
We start with the graph of y = ln x.

32. NATURAL LOGARITHMS
Example 11
Using the transformations of Section 1.3, we shift it 2 units to the right—to get the graph of y = ln(x - 2).

33. NATURAL LOGARITHMS
Example 11
Then, we shift it 1 unit downward—to get the graph of y = ln(x - 2) -1.

34. Here, you can see that the sine function y = sin x is not one-to-one.
INVERSE TRIGONOMETRIC FUNCTIONS
Here, you can see that the sine function
y = sin x is not one-to-one.
Use the Horizontal Line Test.

35. However, here, you can see that the function f(x) = sin x, ,
INVERSE TRIGONOMETRIC FUNCTIONS
However, here, you can see that
the function f(x) = sin x, ,
is one-to-one.

36. INVERSE SINE FUNCTIONS
The inverse sine function, denoted by sin-1 or arcsin, has domain [-1, 1] and range
Its graph is shown.

37. The inverse cosine function is handled similarly.
INVERSE COSINE FUNCTIONS
The inverse cosine function is handled similarly.
The restricted cosine function f(x) = cos x, 0 ≤ x ≤ π, is one-to-one.
So, it has an inverse function denoted by cos-1 or arccos.
The inverse cosine function,cos-1, has domain [-1, 1] and range
Its graph is shown.

38. The tangent function can be made
INVERSE TANGENT FUNCTIONS
The tangent function can be made
one-to-one by restricting it to the interval

39. Thus, the inverse tangent function is defined as
INVERSE TANGENT FUNCTIONS
Thus, the inverse tangent
function is defined as
the inverse of the function
f(x) = tan x, .
It is denoted by tan-1 or arctan.

40. The inverse tangent function, tan-1 = arctan, has domain and range .
INVERSE TANGENT FUNCTIONS
The inverse tangent function, tan-1 = arctan,
has domain and range
Its graph is shown.