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Floor And Ceiling Function
Floor function and ceiling function help in consolidating the function output as an integral value. The floor function gives an integer number value which is a numeric value lesser than the value of the function, and a ceiling function gives an integer number value which is a numeric value greater than the value of the function.
Let us check out the graph of floor and ceiling function, their properties, differences, with the help of examples, FAQs.
What Is Floor Function And Ceiling Function?
The floor function or a ceiling function helps in consolidating the value of a function into a simple integer number value. This helps in simplifying a numeric value in decimal into a simple integer number value.
Floor Function: It is a function that takes an input as a real number and gives an output that is an integral value less than the input real number. The floor function gives the greatest integer output which is lesser than or equal to a given number. The floor function is denoted by floor(x) or \(\lfloor x \rfloor\). Also sometimes the floor function is represented using double brackets and is written as [[x]]. An example of floor function is \(\lfloor 2.3 \rfloor\) = 2, and \(\lfloor 3.4 \rfloor \) = 4.
Ceiling Function: It is a function that takes an input as a real number and gives an output that is an integral value greater than the input real number. The ceiling function gives the least integer output which is greater than or equal to the given number. The ceiling function is denoted by ceil(x) or \(\lceil x \rceil\). Further, the ceiling function is sometimes written using reverse brackets or ]] x [[ or ] x [. An example of a ceiling function is \(\lceil 4.7 \rceil\) = 5, or \(\lceil 2.8 \rceil\) = 2.
Graph Of Floor Function And Ceiling Function
The graph of a floor function or a ceiling function can be taken as a step function or a break function, which are lines parallel to the xaxis. The range of input values is represented as a line and the final floor or the ceiling value is represented as a dot, in the graph of the floor function or the ceiling function.
The floor function for a range of values gives the output for only a minimum integer number value which is represented by a darkened dot. The ceiling function also takes a range of values as input but gives an output which is a maximum integer number value and is represented by a darkened dot.
Properties Of Floor Function And Ceiling Function
The following are some of the important properties of a floor function and ceiling function.
 The floor function and the ceiling function give the output as an integral value.
 The floor function has an integer number value less than the domain value of the function.
 The ceiling function has an integer number value greater than the domain value of the function.
 The floor function can be represented as \(\lfloor x \rfloor \) = max {m ∈ Z  m < x}, and the ceiling function is represented as \(\lceil x \rceil\) = min{n ∈ Z  n > x}.
 If \(\lfloor x \rfloor\) = m then we have m < x < m + 1 and if \(\lceil x \rceil \) = n then n  1 < x < n.
 If \(\lfloor x \rfloor\) = m then x  1 < m < x, and if \(\lceil x \rceil\) = n, then x < n < x + 1.
 For a floor function n < x if and only if n < \(\lfloor x \rfloor \).and for a ceiling function x < n, if and only if we have \(\lceil x \rceil\) < n.
 For a floor function n < x, if and only if n < \(\lfloor x \rfloor\), and for a ceiling function x < n if and only if \(\lceil x \rceil\) < n.
 The sum of a floor function and an integer can be represented as \(\lfloor x + n\rfloor\) = \(\lfloor x \rfloor\) + n, and the sum of a ceiling function and an integer can be represented as \(\lceil x + n \rceil\) = \(\lceil x \rceil\) + n.
 The sum of two floor functions can be represented as \(\lfloor x \rfloor \) + \(\lfloor y \rfloor\) < \(\lfloor x + y \rfloor \) < \(\lfloor x \rfloor \) + \(\lfloor y \rfloor\) + 1, and the sum of two ceiling functions can be represented as \(\lceil x \rceil \) + \(\lceil y \rceil \)  1 < \(\lceil x + y\rceil \) < \(\lceil x \rceil \) + \(\lceil y \rceil\).
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Examples on Floor Function And Ceiling Function

Example 1: What is the value for the floor function f(x) = \(\lfloor 2x + 1 \rfloor \), given that x = 2.8?
Solution:
The given floor function is as follows.
f(x) = \(\lfloor 2x + 1 \rfloor \)
Let us substitute x = 2.8
f(2.8) = \(\lfloor 2(2.8) + 1 \rfloor \)
f(2.8) = \(\lfloor 5.6 + 1 \rfloor \)
f(2.8) = \(\lfloor 6.6 \rfloor \)
f(2.8) = 6
Therefore the value of the floor function is 6.

Example 2: What is the value for the ceiling function for f(x) = \(\lceil 3x  5 \rceil\) given that x = 3.9?
Solution:
The given ceiling function is as follows.
f(x) = \(\lceil 3x  5 \rceil\)
f(3.9) = \(\lceil 3(3.9)  5 \rceil \)
f(3.9) = \(\lceil 11.7  5 \rceil \)
f(3.9) = \(\lceil 6.7 \rceil \)
f(3.9) = 7
Therefore the value of the ceiling function is 7.
FAQs on Floor Function And Ceiling Function
What Is Floor And Ceiling Function in Maths?
The floor function and ceiling function gives an integer value for the decimal answer values of the function. The floor function gives a maximum value that is lesser than or equal to the functional numeric value, and the ceiling function gives a minimum value that is greater than or equal to the functional numeric value.
What Is The Notation Used To Write A Floor Function And Ceiling Function?
The floor function is represented as f(x) = \(\lfloor x \rfloor\), or as [[x]]. And the ceiling function is represented as f(x) = \(\lceil x \rceil \) or as ]] x [[.
What Is The Domain Of Floor Function And Ceiling Function?
The domain of floor function and the ceiling function can be any real number value, which can include a decimal or a fraction value.
What Is The Range Of Floor And Ceiling Function?
The range of a floor function and a ceiling function can be any integer value. The integral value can be a positive value or a negative integer value.
What Is The Difference Between Floor And Ceiling Function?
The floor function gives a value which is lesser than the f(x) value, and the ceiling function gives a value which is greater than the f(x). For a floor function we have \(\lfloor x \rfloor \) < x and for the ceiling function we have x < \(\lceil x \rceil \).
What Is The Target Of Floor Function And Ceiling Functions?
The target of a floor function is an integral value that is lesser than the value of the function, and the target of a ceiling function is an integral value that is greater than the value of the function.
How To Measure Floor Function And Ceiling Function?
The measure of the floor function and the ceiling function is based on the output value of the function. For a function x = 5.6, we have the floor function value of \(\lfloor x \rfloor \) = 5, and the ceiling function value as \(\lceil x \rceil \) = 6.
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