Floor and Ceiling Functions

Question 1

Describe the graph of the floor function 𝑓(𝑥)=⌊𝑥⌋. How does it behave as 𝑥 increases?

Answer 1

The graph of f(x) = ⌊x⌋ consists of a series of horizontal line segments. Each segment is at the integer value n and extends from x = n to x = n + 1, but does not include x = n + 1. At each integer n, there is a jump discontinuity as the function value drops to n.

Question 2

How does the graph of the ceiling function
f(x)=⌈x⌉ differ from the floor function graph?

Answer 2

The graph of f(x) = ⌈x⌉ also consists of horizontal line segments, but each segment is at the integer value n and extends from
x = n − 1 to x = n, but does not include
x = n. There is a jump discontinuity at each integer n where the function value jumps to
n from n − 1.

Question 3

At which points does the ceiling function
f(x)=⌈x⌉ experience discontinuities?

Answer 3

The ceiling function f(x) = ⌈x⌉ experiences discontinuities at every integer value of
x. At each integer n, there is a jump from
n−1 to n.

Question 4

How does the behavior of the floor and ceiling functions differ at integer points?

Answer 4

At integer points, the floor function
⌊x⌋ is constant on intervals that end at these points, and there is no jump at integers. The ceiling function ⌈x⌉ jumps to the next integer at integer points, creating discontinuities at these values.

Question 5

Express the floor function
f(x)=⌊x⌋ and the ceiling function
f(x)=⌈x⌉ as piecewise functions.

Answer 5

Floor Function:
⌊x⌋= n for n ≤ x< n + 1.
Ceiling Function:
⌈x⌉=n for n −1 < x ≤ n.

Question 6

What is the relationship between ⌈x⌉ and ⌊x⌋ for any x?

Answer 6

⌈x⌉ - ⌊x⌋ = either 0 (if x is an integer) or 1 (if x is not an integer)

Question 7

For a real number x, if x lies between n and n + 1, what can be said about ⌈x²⌉ and ⌊x²⌋?

Answer 7

⌊x²⌋ is n² and ⌈x²⌉ is (n + 1)² is x² is not an integer.

Question 8

What is the inequality involving ⌈x⌉ and ⌊x⌋?

Answer 8

⌊x⌋ ≤ x < ⌈x⌉ always holds true.