The Greatest and Least Integer Functions


For what values of x is


b. ⌈x⌉ = 0

Verified step by step guidance

1. Understand the ceiling function ⌈x⌉, which rounds a number up to the nearest integer. For example, ⌈2.3⌉ = 3 and ⌈-1.7⌉ = -1.
2. To find the values of x for which ⌈x⌉ = 0, consider the definition: ⌈x⌉ is the smallest integer greater than or equal to x.
3. Since ⌈x⌉ = 0, x must be less than or equal to 0 but greater than -1, because ⌈x⌉ rounds up to the nearest integer.
4. Therefore, the values of x that satisfy ⌈x⌉ = 0 are those in the interval (-1, 0].
5. Express the solution in interval notation: x ∈ (-1, 0].

Key Concepts

Greatest Integer Function (Ceiling Function)

The greatest integer function, denoted as ⌈x⌉, returns the smallest integer that is greater than or equal to x. For example, ⌈2.3⌉ equals 3, while ⌈-1.5⌉ equals -1. This function is crucial for understanding how to manipulate and solve equations involving integer constraints.

Understanding Zero in the Context of the Ceiling Function

To solve the equation ⌈x⌉ = 0, we need to determine the range of x values that yield a ceiling of zero. Since the ceiling function rounds up to the nearest integer, this means x must be in the interval [-1, 0). Thus, any value of x within this range will satisfy the equation.

Intervals and Inequalities

Understanding intervals and inequalities is essential for solving equations involving functions like the ceiling function. In this case, the solution involves identifying the interval where the function holds true, which is critical for determining valid x values that meet the specified condition.