Ceiling Function

Ceiling Function is an important function in mathematics that returns the smallest integer which is not smaller than the input decimal. It is usually expressed as a function of a variable and denoted either by f(x) or by ceil(x) or ⌈x⌉. Ceiling Function has applications in various fields such as physics, electronics, and AI due to which it becomes much more important to study ceiling function.

A ceiling function is neither a one-one nor an onto function as various elements have the same image and a pre-image has various images in the co-domain and domain set respectively. In this article, we shall discuss the ceiling function in detail.

What is Ceiling Function?

Ceiling Function is a special type of function in Mathematics that returns the smallest integer which is not smaller than the input decimal. If we input a negative decimal in ceiling function then the result will be the whole integer value of that decimal whereas If we input a positive decimal in ceiling function then the result will be 1 more than the whole integer value of that decimal. In the case of zero, the output is always zero.

Ceiling Function Definition

Mathematically, the ceiling function is defined as follows:

Ceil(x) OR ⌈x⌉ = min{n ∈ Z : n ≥ x}

This means that the ceiling function returns the least integer that is greater than or equal to x i.e. the input number.

Ceiling Function Symbol

Ceiling Function is denoted using the symbol ⌈⌉. Thus we can denote ceiling(x) by ⌈x⌉. Other than this, ceiling function is also denoted by abbrivation of ceiling i.e., ceil(x).

Domain And Range Of Ceiling Function

The Domain of the ceiling function is all real numbers i.e. R and the co-domain and range of the signum function are set of all integers i.e. Z.

Graph Of Ceiling Function

The graph of a ceiling function is a step graph or a broken graph in which the plotted lines are parallel to the X-axis. On the graph, a line represents the range of inputs and the output of ceiling function is shown using a circle. The maximum integer value returned by ceiling function is shown by a dot. The graph of ceiling function is shown below:

Properties Of Ceiling Function

Properties of Ceiling function are used to simplify the equations that involve the use of Ceiling function. Ceiling Function ⌈x⌉ has the following properties:

• The value returned by ⌈x⌉ is always an integer.
• If ⌈x⌉ = a then a-1 < x ≤ a
• If ⌈x⌉ = a then x ≤ a < x + 1
• ⌈x⌉ + ⌈y⌉ – 1 ≤ ⌈x + y⌉ ≤ ⌈x⌉ + ⌈y⌉
• ⌈x + a⌉ = ⌈x⌉ + a
• a < ⌈x⌉ if a < x
• a ≤ ⌈x⌉ if x < a

Floor And Ceiling Function

Floor function is a function which returns the greatest integer which is smaller than or equal to the input number. It is represented using ⌊x⌋, where x is the input number. Difference between floor and ceiling function are as follows:

Ceiling Function	Floor Function
It returns the smallest integer greater than or equal to the integer of the input number.	It returns the greatest integer smaller than or equal to the integer of the input number.
If the input x is a positive decimal, then ceiling function returns 1 more than whole integer part of the decimal.	If the input x is a positive decimal, then floor function returns the integer part of the decimal.
If input x is a negative decimal number, then the output is integer part of the decimal.	If input x is a negative decimal number, then output is 1 less than the integer part of the decimal.
It is represented using ⌈x⌉.	It is represented using ⌊x⌋.

Applications Of Ceiling Function

Ceiling function has various applications in different fields. Some of its applications are:

• Calculating the value of postage stamps makes use of ceiling function.
• It is also used by various companies to calculate the billing amount.
• It is used to solve complex problems in mathematics, science and engineering.
• It is used to find the smallest number which is greater than or equal to a given number.

Solved Examples On Ceiling Function

Example 1: Find possible values of x If ⌈x⌉ = 4.

Solution:

We know that ceiling function returns 1 more than the input decimal if the input is positive.

Thus we can say that in this case value of x will be greater than 3 but less than or equal to 4.

3 < x ≤ 4

Example 2: Find the possible values of x If ⌈x⌉ = -3.

Solution:

We know that ceiling function returns the whole integer value of input decimal if the input is negative.

Thus we can say that in this case value of x will be less than or equal to -3.

-3 > x

Example 3: Calculate the value of the ceiling function for the values in the set [1.3, -0.51, 0.465, 1].

Solution:

We know that

⌈1.3⌉ = 2

⌈-0.51⌉ = 0

⌈0.465⌉ = 1

⌈1⌉ = 1

Example 4: Calculate the value of ⌈5.1 + 1⌉.

Solution:

We know that ⌈x + a⌉ = ⌈x⌉ + a

⌈5.1 + 1⌉ = ⌈5.1⌉ + 1 = 6+1

= 7

Example 5: Calculate the value of the ceiling function for the values in the set [-0.3, -0.91, 3.465, -9.4].

Solution:

We know that

⌈-0.3⌉ = 0

⌈-0.91⌉ = 0

⌈3.465⌉ = 4

⌈-9.4⌉ = -9

Practice Problems on Ceiling Function

Q1: What is the value of ⌈6.7⌉?

Q2: Calculate ⌈-3.4⌉.

Q3: Determine ⌈2.71828⌉ (where 2.71828 is the value of the mathematical constant “e”).

Q4: If x is an even positive integer, express ⌈x/2⌉ in terms of x.

Q5: Solve for x in the equation ⌈3x – 2⌉ = 9.

Q6: Given a real number y, find the largest integer n such that ⌈y⌉ = n.

Q7: Compute ⌈⌈5.5⌉ + ⌈3.9⌉⌉.

Q8: What is the sum of the first 4 positive integers rounded up to the nearest integer using the ceiling function?

Q9: Determine the value of ⌈⌈⌈8.2⌉/4⌉/2⌉.

Q10: Solve for x in the equation ⌈1.5x⌉ = 6.

FAQs On Ceiling Function

1. Define Ceiling Function?

Ceiling function is a special type of function in Mathematics that returns the smallest integer which is not smaller than the input decimal.

2. Is Ceiling Function An Even Function Or Odd Function?

Ceiling function is neither an odd function nor an even function as

⌈x⌉ ≠ ⌈-x⌉

and

⌈x⌉ ≠ -⌈-x⌉

3. Give Some Applications Of Ceiling Function?

Some applications of ceiling function are:

• Calculating the value of postage stamps makes use of ceiling function.
• It is also used by various companies to calculate the billing amount.

4. Give The Domain Of Ceiling Function?

Domain of ceiling function is the set of all real numbers i.e. R.

5. How Does The Graph Of Ceiling Function Look Like?

The graph of a ceiling function is a graph like the step function and contains of parallel lines, circles and dots representing range of inputs, and the output of ceiling function.

6. What is an Example of Ceiling Function?

An example of ceiling function is ⌈4.1⌉ = 5.

7. What Is The Floor(4.2) and Ceil(2.9) Equal To?

Floor(4.2) is equal to 4 and ceiling(2.9) is equal to 3.

8. Why Ceiling Function Is Called Step Function?

Ceiling Function is called a step function as the graph of ceiling function resembles the steps of stairs. The ceiling function also does not produce continuous graph and is broken into a series of steps.