1. The problem is to understand the meaning and properties of the ceiling function $\lceil x \rceil$. 2. The ceiling function $\lceil x \rceil$ is defined as the smallest integer greater than or equal to $x$. 3. For example, if $x = 2.3$, then $\lceil 2.3 \rceil = 3$ because 3 is the smallest integer not less than 2.3. 4. If $x$ is already an integer, say $x = 5$, then $\lceil 5 \rceil = 5$. 5. This function is useful in rounding numbers up to the nearest integer. 6. Important properties: - $\lceil x \rceil \geq x$ - $\lceil x \rceil$ is always an integer - If $x$ is an integer, $\lceil x \rceil = x$ 7. To evaluate $\lceil x \rceil$ for any real number $x$, find the smallest integer $n$ such that $n \geq x$. Final answer: The ceiling function $\lceil x \rceil$ rounds $x$ up to the nearest integer.