1. Today in Pre-Calculus Go over homework
Notes: Determining if a function is bounded below, bounded above or unbounded
- need a calculator
Homework

2. incr: (- ∞, ∞)
decr: (- ∞, 0 )
incr: (0, ∞)
decr: (- ∞, 0 )
incr: (0, ∞)
decr: ( 3, 5 )
incr: (-∞, 3 )
constant: ( 5, ∞)
decr: ( 3, ∞)
incr: (-∞, 0 )
constant: (0, 3)
decr: (- 1, 1)
incr: (- ∞, -1 ), ( 1, ∞)
decr: (- ∞, ∞)
incr: (- ∞, 0)
decr: (0, ∞)
decr: (- ∞, -4)
incr: ( 4, ∞)
decr: (2,∞)
incr: (-∞,-2)
constant(-2,2)
Inc(0,3)
decr: (- ∞, 0)
cons: (3, ∞)
decr: ( - ∞, 7)υ (7, ∞)

3. Functions Bounded Below
Definition: A function f is bounded below if there is some number b that is less than or equal to every number in the range of f.
Answers is in terms of y-values
Any such number b is called a lower bound of f.
In this graph b=-2

4. Functions Bounded Above
A function f is bounded above if there is some number B that is greater than or equal to every number in the range of f.
Any such number B is called an upper bound of f.
In this graph B = 3

5. Bounded
A function f is bounded if it is bounded from both above and below.
In this graph b = -1 and B = 1.

6. Unbounded
A function f is unbounded if it is neither bounded from above and below.
As separate pieces (or branches), the lower piece is bounded above and the upper piece is bounded below, however as a whole the function f is unbounded.

7. Example 1 Bounded below b = 3 Prove Algebraically: x2≥0 2x2≥0

8. Example 2
Bounded
b = -1
B = 1

9. Example 3
Bounded above
B = 5

10. Example 4
unbounded

11. Homework
Wkst.