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# Points, Vectors, and Functions

# Bounded Exercises

### 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*) =*x*^{2}

### 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.