1. College Algebra Fifth Edition
James Stewart  Lothar Redlin  Saleem Watson

2. 3
Functions

3. One-to-One Functions and Their Inverses
3.7

4. Inverse of a Function
The inverse of a function is a rule that acts on the output of the function and produces the corresponding input.
So, the inverse “undoes” or reverses what the function has done.

5. Not all functions have inverses.
One-to-One Function
Not all functions have inverses.
Those that do are called one-to-one.

6. One-to-One Functions

7. Let’s compare the functions f and g whose arrow diagrams are shown.
One-to-One Functions
Let’s compare the functions f and g whose arrow diagrams are shown.
f never takes on the same value twice (any two numbers in A have different images).
g does take on the same value twice (both 2 and 3 have the same image, 4).

8. In symbols, g(2) = g(3) but f(x1) ≠ f(x2) whenever x1 ≠ x2
One-to-One Functions
In symbols, g(2) = g(3) but f(x1) ≠ f(x2) whenever x1 ≠ x2
Functions that have this latter property are called one-to-one.

9. One-to-One Function—Definition
A function with domain A is called a one-to-one function if no two elements of A have the same image, that is, f(x1) ≠ f(x2) whenever x1 ≠ x2

10. One-to-One Functions
An equivalent way of writing the condition for a one-to-one function is as follows: If f(x1) = f(x2), then x1 = x2.

11. One-to-One Functions
If a horizontal line intersects the graph of f at more than one point, then we see from the figure that there are numbers x1 ≠ x2 such that f(x1) = f(x2).

12. This means that f is not one-to-one.
One-to-One Functions
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.

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

14. E.g. 1—Deciding Whether a Function Is One-to-One
Is the function f(x) = x3 one-to-one?

15. E.g. 1—If a Function Is One-to-One
Solution 1
If x1 ≠ x2, then x13 ≠ x23.
Two different numbers cannot have the same cube.
Therefore, f(x) = x3 is one-to-one.

16. E.g. 1—If a Function Is One-to-One
Solution 2
From the figure, we see that no horizontal line intersects the graph of f(x) = x3 more than once.
Hence, by the Horizontal Line Test, f is one-to-one.

17. One-to-One Functions
Notice that the function f of Example 1 is increasing and is also one-to-one.
In fact, it can be proved that every increasing function and every decreasing function is one-to-one.

18. E.g. 2—If a Function Is One-to-One
Solution 1
Is the function g(x) = x2 one-to-one?
This function is not one-to-one because, for instance, g(1) = 1 and g(–1) = 1 and so 1 and –1 have the same image.

19. E.g. 2—If a Function Is One-to-One
Solution 2
From the figure, we see that there are horizontal lines that intersect the graph of g more than once.
So, by the Horizontal Line Test, g is not one-to-one.

20. One-to-One Functions
Although the function g in Example 2 is not one-to-one, it is possible to restrict its domain so that the resulting function is one-to-one.

21. One-to-One Functions
In fact, if we define h(x) = x2 , x ≥ 0 then h is one-to-one, as you can see from the figure and the Horizontal Line Test.

22. E.g. 3—Showing that a Function Is One-to-One
Show that the function f(x) = 3x + 4 is one-to-one.
Suppose there are numbers x1 and x2 such that f(x1) = f(x2).
Then,
Therefore, f is one-to-one.

23. The Inverse of a Function

24. Inverse of a Function
One-to-one functions are important because they are precisely the functions that possess inverse functions according to the following definition.

25. Inverse of a Function—Definition
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.

26. Inverse of a Function
This definition says that, if f takes x into y, then f –1 takes y back into x.
If f were not one-to-one, then f –1 would not be defined uniquely.

27. This arrow diagram indicates that f –1 reverses the effect of f.
Inverse of a Function
This arrow diagram indicates that f –1 reverses the effect of f.

28. Inverse of a Function
From the definition, we have: domain of f –1 = range of f range of f –1 = domain of f

29. Don’t mistake the –1 in f –1 for an exponent.
Warning on Notation
Don’t mistake the –1 in f –1 for an exponent.
f –1 does not mean 1/f(x)
The reciprocal 1/f(x) is written as (f(x))–1.

30. E.g. 4—Finding f –1 for Specific Values
If f(1) = 5, f(3) = 7, f(8) = –10 find f –1(5), f –1(7), f –1(–10)

31. E.g. 4—Finding f –1 for Specific Values
From the definition of f –1, we have: f –1(5) = because f(1) = f –1(7) = because f(3) = f –1(–10) = because f(8) = –10

32. E.g. 4—Finding f –1 for Specific Values
The figure shows how f –1 reverses the effect of f in this case.

33. The Inverse of a Function
By definition, the inverse function f –1 undoes what f does:
If we start with x, apply f, and then apply f –1, we arrive back at x, where we started.
Similarly, f undoes what f –1 does.

34. The Inverse of a Function
In general, any function that reverses the effect of f in this way must be the inverse of f.
These observations are expressed precisely—as follows.

35. Inverse Function Property
Let f be a one-to-one function with domain A and range B.
The inverse function f –1 satisfies the following cancellation properties.
Conversely, any function f –1 satisfying these equations is the inverse of f.

36. Inverse Function Property
These properties indicate that f is the inverse function of f –1.
So, we say that f and f –1 are inverses of each other.

37. E.g. 5—Verifying that Two Functions Are Inverses
Show that f(x) = x3 and g(x) = x1/3 are inverses of each other.
Note that the domain and range of both f and g is .
We have:
So, by the Property of Inverse Functions, f and g are inverses of each other.

38. E.g. 5—Verifying that Two Functions Are Inverses
These equations simply say that:
The cube function and the cube root function, when composed, cancel each other.

39. Computing Inverse Functions
Now, let’s examine how we compute inverse functions.
First, we observe from the definition of f –1 that:

40. Computing Inverse Functions
So, if y = f(x) and if we are able to solve this equation for x in terms of y, then we must have x = f –1(y).
If we then interchange x and y, we have y = f –1(x) which is the desired equation.

41. How to Find the Inverse of a One-to-One Function
Write y = f(x).
Solve this equation for x in terms of y (if possible).
Interchange x and y. The resulting equation is y = f –1(x)

42. How to Find the Inverse of a One-to-One Function
Note that Steps 2 and 3 can be reversed.
That is, we can interchange x and y first and then solve for y in terms of x.

43. E.g. 6—Finding the Inverse of a Function
Find the inverse of the function f(x) = 3x – 2
First, we write y = f(x): y = 3x – 2
Then, we solve this equation for x:

44. E.g. 6—Finding the Inverse of a Function
Finally, we interchange x and y:
Therefore, the inverse function is:

45. E.g. 7—Finding the Inverse of a Function
Find the inverse of the function
We first write y = (x5 – 3)/2 and solve for x.

46. E.g. 7—Finding the Inverse of a Function
Then, we interchange x and y to get: y = (2x + 3)1/5
Therefore, the inverse function is: f –1(x) = (2x + 3)1/5

47. Graphing the Inverse of a Function
The principle of interchanging x and y to find the inverse function also gives a method for obtaining the graph of f –1 from the graph of f.
If f(a) = b, then f –1(b) = a.
Thus, 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.

48. Graphing the Inverse of a Function
However, we get the point (b, a) from the point (a, b) by reflecting in the line y = x.

49. Graph of the Inverse Function
Therefore, as this figure illustrates, the following is true:
The graph of f –1 is obtained by reflecting the graph of f in the line y = x.

50. E.g. 8—Finding the Inverse of a Function
Sketch the graph of
Use the graph of f to sketch the graph of f –1.
Find an equation for f –1.

51. E.g. 8—Finding Inverse of Function
Example (a)
Using the transformations from Section 3.5, we sketch the graph of by:
Plotting the graph of the function
Moving it to the right 2 units.

52. E.g. 8—Finding Inverse of Function
Example (b)
The graph of f –1 is obtained from the graph of f in part (a) by reflecting it in the line y = x.

53. E.g. 8—Finding Inverse of Function
Example (c)
Solve for x, noting that y ≥ 0.

54. E.g. 8—Finding Inverse of Function
Example (c)
Interchange x and y:
y = x2 + 2, x ≥ 0 Thus, f –1(x) = x2 + 2, x ≥ 0
This expression shows that the graph of f –1 is the right half of the parabola y = x2 + 2.

55. E.g. 8—Finding Inverse of Function
Example (c)
From the graph shown in the figure, that seems reasonable.