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Points, Vectors, and Functions

Points, Vectors, and Functions

Example 1

Exercises. Determine if each function is (a) bounded above (b) bounded below.

  • y = cos x

Example 2

Exercises. Determine if each function is (a) bounded above (b) bounded below.

  • f(x) = 5

Example 3

Exercises. Determine if each function is (a) bounded above (b) bounded below.

  • f(x) = x2

Example 4

Exercises. Determine if each function is (a) bounded above (b) bounded below.

  • y = x

Example 5

Exercises. Determine if each function is (a) bounded above (b) bounded below.

Example 6

Exercises. Determine if each function is (a) bounded above (b) bounded below.

Example 7

Exercises. Determine if each function is (a) bounded above (b) bounded below.

Example 8

Exercises. Determine if each function is (a) bounded above (b) bounded below.

Example 9

Exercises. Determine if each function is (a) bounded above (b) bounded below.

Example 10

Exercises. Determine if each statement is true or false. 

  • If a function is bounded above it must also be bounded below.

Example 11

Exercises. Determine if each statement is true or false. 

  • If a function f(x) has an upper bound of M there must be some value of x for which f(x) = M.

Example 12

Exercises. Determine if each statement is true or false. 

  • If there is some K such that f(x) ≥ K for all x, then the function f(x) is bounded below. 
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