Today in Pre-Calculus

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. decr: (- ∞, 0 ) incr: (0, ∞) incr: (- ∞, ∞) decr: (- ∞, 0 ) incr: (0, ∞) decr: ( 3, ∞) incr: (-∞, 0 ) constant: (0, 3) decr: ( 3, 5 ) incr: (-∞, 3 ) constant: ( 5, ∞) decr: (- 1, 1) incr: (- ∞, -1 ), ( 1, ∞) decr: (- ∞, ∞) incr: (- ∞, 0) decr: (0, ∞) decr: (2,∞) incr: (-∞,-2) constant(-2,2) decr: (- ∞, -4) incr: ( 4, ∞) 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 2x2+3≥0+3 2x2+3≥3

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

9. Example 3 Bounded above B = 5

10. Example 4 unbounded

11. Homework • Wkst.