Ceiling Function

1. Fundamental Concepts
Definition:
The ceiling function, denoted as $\lceil x \rceil$, returns the smallest integer greater than or equal to $x$.

Examples:
$$\lceil 3.2 \rceil = 4, \quad \lceil -2.7 \rceil = -2, \quad \lceil 5 \rceil = 5$$

Graphical Representation:
The graph consists of horizontal line segments, similar to the floor function, but each step “jumps upward.” It is discontinuous at integer points.

2. Key Concepts
Evaluation Rule:
$$\lceil x \rceil = n \quad \text{if} \quad n - 1 < x \leq n$$
Relation to Floor Function:
$$\lceil x \rceil = -\lfloor -x \rfloor$$
(This is often the easiest way to compute ceilings.)

Applications:
Used in computer science for memory allocation or rounding up divisions. Useful in scheduling problems where fractional tasks need to be rounded up to whole units.

3. Examples
Example 1 (Positive Number):
Evaluate $\lceil 4.3 \rceil$.
The smallest integer ≥ 4.3 is 5. So, $\lceil 4.3 \rceil = 5$.

Example 2 (Negative Number):
Evaluate $\lceil -3.2 \rceil$.
The smallest integer ≥ -3.2 is -3. So, $\lceil -3.2 \rceil = -3$.

4. Problem-Solving Techniques
Number Line Visualization: Locate $x$ on the number line, then round upward to the nearest integer.
Check Boundaries: Ensure the chosen integer satisfies $n-1 < x \leq n$.
Use the Floor Relationship: For tricky negative values, compute using $\lceil x \rceil = -\lfloor -x \rfloor$.