Domain
In biological taxonomy, a domain is the highest rank of classification for living organisms, grouping them into three fundamental categories—Bacteria, Archaea, and Eukarya—based on evolutionary divergences inferred from molecular data such as ribosomal RNA sequences.[1][2] This three-domain system, proposed by Carl Woese and colleagues in 1990, revolutionized phylogeny by elevating domains above kingdoms to better reflect monophyletic clades derived from genetic comparisons rather than morphological traits alone.[1]/Unit_1:_Introduction_to_Microbiology_and_Prokaryotic_Cell_Anatomy/1:_Fundamentals_of_Microbiology/1.3:Classification-_The_Three_Domain_System) The framework highlights Archaea's distinctiveness from Bacteria—despite both being prokaryotic—through differences in membrane lipids, DNA replication, and protein synthesis, enabling Archaea to thrive in extreme conditions like high temperatures and acidity.[3][4] Eukarya, encompassing protists, fungi, plants, and animals, is characterized by membrane-bound nuclei and organelles, tracing its origins to endosymbiotic events involving bacterial ancestors.[5] While the system has faced debate over the universality of rRNA trees and potential for additional domains like viruses or organelles, it remains the prevailing model due to corroboration from genomic and biochemical evidence.[2][6]Mathematics
Domain of a function
In mathematics, the domain of a function f is the set of all possible input values for which the function produces a defined output.[7] For a function f: X \to Y, the domain is the set X, comprising elements that can be mapped to elements in the codomain Y.[8] This concept ensures the function is well-defined, avoiding operations like division by zero or taking even roots of negative numbers in real-valued contexts.[9] In set theory, a function is a special relation—a subset of the Cartesian product X \times Y where each element of X appears in exactly one ordered pair. The domain is then the set of all first components of those pairs, formally \{x \in X \mid \exists y \in Y \text{ such that } (x, y) \in f\}.[10] This foundational view underpins abstract algebra and topology, where domains may be arbitrary sets rather than subsets of real numbers. For practical purposes, such as in calculus, the domain of a real-valued function defined by a formula is its natural domain: the largest subset of \mathbb{R} (or \mathbb{R}^n) where the expression yields real outputs without undefined operations.[11] To determine the domain of an algebraic function, identify and exclude inputs causing indeterminacies:- Denominators must not equal zero; for f(x) = \frac{1}{x-2}, solve x-2 \neq 0, so domain is \mathbb{R} \setminus \{2\}.[12]
- Even-indexed roots require non-negative arguments; for f(x) = \sqrt{x+3}, x+3 \geq 0 implies x \geq -3, so domain is [-3, \infty).[13]
- Logarithms demand positive arguments; for f(x) = \log_2(x-1), x-1 > 0 yields x > 1, domain (1, \infty).[9] Compositions or piecewise definitions may impose combined restrictions; for f(x) = \frac{\sqrt{x}}{x^2 - 4}, numerator requires x \geq 0 and denominator x \neq \pm 2, so domain is [0, \infty) \setminus \{2\}.[14]