Today in PreCalculus Go over homework Notes Determining

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

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

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

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

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

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.

Example 1 Bounded below b=3 Prove Algebraically: x 2≥ 0 2 x 2+3≥ 0+3 2 x 2+3≥ 3

Example 2 Bounded b = -1 B=1

Example 3 Bounded above B=5

Example 4 unbounded

Homework • Wkst.