Ceiling Function

Definition, Graph, Evaluations, and Properties

Tutorial on the ceiling function, including graphing and evaluations, is presented.

Definition of the Ceiling Function

The ceiling function \( \lceil x \rceil \) (or \( \text{ceiling}(x) \)) is defined as the function that outputs the smallest integer greater than or equal to \( x \). Below is shown the graph of \( \text{ceiling}(x) \):

graph of the ceiling function

The domain of \( \text{ceiling}(x) \) is the set of all real numbers. The range of \( \text{ceiling}(x) \) is the set of all integers.

Evaluations and Examples

Evaluate \( \text{ceiling}(x) \) for various values of \( x \).

Example 1

Evaluate \( \text{ceiling}(1.5) \)

Solution
\[ \text{ceiling}(1.5) = 2 \]

Example 2

Evaluate \( \text{ceiling}(0) \)

Solution
\[ \text{ceiling}(0) = 0 \]

Example 3

Evaluate \( \text{ceiling}(2) \)

Solution
\[ \text{ceiling}(2) = 2 \]

Example 4

Evaluate \( \text{ceiling}(-3) \)

Solution
\[ \text{ceiling}(-3) = -3 \]

Example 5

Evaluate \( \text{ceiling}(-1.5) \)

Solution
\[ \text{ceiling}(-1.5) = -1 \]

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