# Smallest Integer Function (Ceiling Function) – Graph, Domain and Range

Here you will learn what is smallest integer function definition, graph and its domain and range with examples.

Let’s begin –

## Smallest Integer Function (Ceiling Function)

Definition : For any real number x, we use the symbol $$\lceil x \rceil$$ to denote the smallest integer greater than or equal to x.

For example, $$\lceil 4.7 \rceil$$ = 5, $$\lceil -7.2 \rceil$$ = -7, $$\lceil 5 \rceil$$ = 5, $$\lceil 0.75 \rceil$$ = 1 etc.

The function f : R $$\rightarrow$$ R defined by f(x) = $$\lceil x \rceil$$ for all x $$\in$$ R is callled the smallest integer function or the ceiling function.

It is also a step function.

## Graph of Ceiling Function

The graph of the smallest integer function is given below.

## Domain and Range of Smallest Integer Function

We observe that the domain of the smallest integer function is the set R of all real numbers and its range is the set Z  of all integers.

Domain : R

Range : Z

## Properties :

Following are some properties of smallest integer function.

(i) $$\lceil -n \rceil$$ = – $$\lceil n \rceil$$ where n $$\in$$ Z.

(ii) $$\lceil -x \rceil$$ = – $$\lceil x \rceil$$ + 1, where x $$\in$$ R – Z

(iii) $$\lceil x + n \rceil$$ = $$\lceil x \rceil$$ + n, where x $$\in$$ R – Z and n $$\in$$ Z

(iv) $$\lceil x \rceil$$ + $$\lceil -x \rceil$$ = {1, if x $$\notin$$ Z and 0, if x $$\in$$ Z}

(v) $$\lceil x \rceil$$ + $$\lceil -x \rceil$$ = {2$$\lceil x \rceil$$ – 1,  if x $$\notin$$ Z  and  2$$\lceil x \rceil$$, if x $$\in$$ Z}.