College Algebra Sixth Edition James Stewart  Lothar Redlin  Saleem Watson

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

2. Functions 2

3. One-to-One Functions and Their Inverses 2.7

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. One-to-One Functions • Not all functions have inverses. • Those that do are called one-to-one.

6. One-to-One Functions

7. 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. 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 fat more than one point, then we see from the figure that there are numbers x1 ≠ x2such that f(x1) = f(x2).

12. 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. 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) = x3one-to-one?

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

16. Solution 2 E.g. 1—If a Function Is One-to-One • 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. Solution 1 E.g. 2—If a Function Is One-to-One • 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. Solution 2 E.g. 2—If a Function Is One-to-One • 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 defineh(x) = x2,x≥ 0then 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 x2such 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 to 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. 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. 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) = 1 because f(1) = 5f–1(7) = 3 because f(3) = 7f–1(–10) = 8 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. 6—Verifying that Two Functions Are Inverses • Show that f(x) = x3 and g(x) = x1/3are inverses of each other. • Note that the domain and range of both f and gis . • We have: • So, by the Property of Inverse Functions, f and g are inverses of each other.

38. E.g. 6—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. Finding the Inverse of a Function • Now, let’s examine how we compute inverse functions. • First, we observe from the definition of f –1that:

40. Finding the Inverse of a Function • 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. 7—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. 7—Finding the Inverse of a Function • Finally, we interchange x and y: • Therefore, the inverse function is:

45. E.g. 8—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. 8—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

48. 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 fif and only if the point (b, a) is on the graph of f –1.

49. 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.

50. 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.