1. Increasing and Decreasing Functions
Section 4.1
Increasing and Decreasing Functions

2. Objectives: 1. To give intervals where a function is
increasing or decreasing using
interval notation.
2. To identify one-to-one functions
using the horizontal line test.
3. To identify monotonic functions.

3. A closed interval is a segment (interval that includes both endpoints) and is represented with square brackets.
An open interval is one that does not include its endpoints and is represented with parentheses.

4. -6 -4 -2 0 2 4 6 Type of Interval Inequality Interval Notation closed
Type of Interval
Inequality
Interval Notation
closed
-2  x  1
[-2, 1]

5. -6 -4 -2 0 2 4 6 Type of Interval Inequality Interval Notation open
Type of Interval
Inequality
Interval Notation
open
1  x  4
(1, 4)

6. -6 -4 -2 0 2 4 6 Type of Interval Inequality Interval Notation
Type of Interval
Inequality
Interval Notation
half-open
0  x  3
[0, 3)

7. EXAMPLE 1 Write the set shown in interval notation.
Answer [-2, ∞)

8. Definition
One-to-one correspondence of sets A pairing of elements of two sets so that any element of either set is paired with exactly one element of the other set.

9. Consider the mapping for f = {(1, 7), (2,5), (3,5), (4,6)}
f is a function, but not one-to-one.

10. Definition
One-to-one function f is one-to-one if and only if f(a) = f(b) implies a = b,  a, b  Df (domain of f ).

11. Consider the mapping for g = {(5, 8),
(-2,3), (4,7)} D 5 -2 4 R 8 3 7
The function g is one-to-one.

12. Definition
Increasing function A function is increasing if and only if for any two points x1 and x2  R, x1  x2 implies f(x1)  f(x2).

13. Definition
Decreasing function A function is decreasing if and only if for any two points x1 and x2  R, x1  x2 implies f(x1)  f(x2).

14. Increasing Function

15. Decreasing Function

16. Decreasing Function x y

17. A function that is increasing over its entire domain or one that is decreasing over its entire domain is a strictly monotonic function.

18. A monotonic function may remain constant over all or part of its domain. It is either nondecreasing or nonincreasing.

19. Definition
Nondecreasing function A function is nondecreasing if and only if for any two points x1 and x2  R, x1  x2 implies f(x1)  f(x2).

20. Definition
Nonincreasing function A function is nonincreasing if and only if for any two points x1 and x2  R, x1  x2 implies f(x1)  f(x2).

21. Increasing and also nondecreasingx y

22. Not increasing but still nondecreasingx y

23. Homework: pp

24. ►A. Exercises Give the interval notation for each graph. 3. 1. (-3, 3)
1. (-3, 3)
2. (-3, 3]
3. [-3, 3)
4. [-3, 3]

25. ►A. Exercises Give the interval notation for each graph. 5. 1. (4, ∞)
1. (4, ∞)
2. (4, ∞]
3. [4, ∞)
4. [4, ∞]

26. ►A. Exercises
9. Use proper interval notation to state where the function is increasing, decreasing, or constant. Tell whether the function is one-to-one. x y

27. ►B. Exercises
Graph the following functions. Being as specific as possible, classify each over its entire domain as increasing, decreasing, nonincreasing, nondecreasing, or none. Tell whether the function is one-to-one.
11. y = x2

28. ►B. Exercises
11. y = x2 x y

29. ►B. Exercises
Graph the following functions. Being as specific as possible, classify each over its entire domain as increasing, decreasing, nonincreasing, nondecreasing, or none. Tell whether the function is one-to-one.
13. y = 2-x

30. ►B. Exercises
13. y = 2-x x y

31. ►B. Exercises
Graph the following functions. Being as specific as possible, classify each over its entire domain as increasing, decreasing, nonincreasing, nondecreasing, or none. Tell whether the function is one-to-one.
15. y = x2 – 3

32. ►B. Exercises
15. y = x2 – 3 x y

33. ►B. Exercises
Graph the following functions. Being as specific as possible, classify each over its entire domain as increasing, decreasing, nonincreasing, nondecreasing, or none. Tell whether the function is one-to-one.
19. y =
x if x  1
x + 2 if x  1   

34. ►B. Exercises
19. y =
x if x  1
x + 2 if x  1    x y

35. ■ Cumulative Review
Without graphing, classify each type of function and evaluate it for x = 3.
27. f(x) = sin x  2

36. ■ Cumulative Review
Without graphing, classify each type of function and evaluate it for x = 3.
28. g(x) = x2 + 4x + 4

37. ■ Cumulative Review
Without graphing, classify each type of function and evaluate it for x = 3.
29. h(x) = 5 ● 2x

38. ■ Cumulative Review
Without graphing, classify each type of function and evaluate it for x = 3.
30. k(x) =
2, x  3
x2, x  3   

39. ■ Cumulative Review
Without graphing, classify each type of function and evaluate it for x = 3.
31. q(x) =
x2 + x – 1
x3 – 2x2 – x + 5