Which Function Is Positive For The Entire Interval 3 2

Which function is positive for the entire interval 3 2?

Understanding whether a mathematical expression stays positive throughout a specified range is a foundational skill in calculus, algebra, and many applied sciences. This question—which function is positive for the entire interval 3 2—invites a systematic exploration of sign analysis, interval notation, and the tools needed to verify positivity across every point between the endpoints. In this article we will dissect the concept, outline the logical steps to identify suitable functions, and provide concrete examples that illustrate the process. By the end, readers will possess a clear roadmap for selecting or constructing functions that remain strictly greater than zero on any closed interval, including the illustrative case of the interval bounded by 2 and 3.

Understanding the Interval Notation

Before tackling the core question, it is essential to clarify the interval itself. In standard mathematical notation, an interval is expressed as [a, b] where a denotes the lower bound and b the upper bound. When the bounds are written as “3 2,” the conventional interpretation is that the interval runs from 2 to 3 (i.e., [2, 3]). The order matters: a reversed notation such as [3, 2] would represent an empty set because the lower limit cannot exceed the upper limit. Consequently, the phrase “the entire interval 3 2” is best understood as referring to the closed interval [2, 3].

Key takeaway: The interval under discussion is the set of all real numbers x such that 2 ≤ x ≤ 3.

Criteria for a Function to Be Positive on an Interval

A function f(x) is said to be positive on an interval [a, b] if, for every x in that interval, *f(x) >