domain

Examples of domain in the following topics:

• Visualizing Domain and Range

  • The domain is part of the definition of a function.  
  • For example, the domain of the function $f(x) = \sqrt{x} $ is $x\geq0$.
  • By definition, a function only has one result for each domain.  
  • Example 1:  Determine the domain and range of each graph pictured below:
  • Use the graph of a function to determine its domain and range

• Restricting Domains to Find Inverses

  • Informally, a restriction of a function is the result of trimming its domain.  
  • Is  $x=0$ in the domain of the function $f(x)=log(x)$?  
  • To verify, suppose $x=0$ is in the domain of the function $f(x)=log(x)$. 
  • Therefore, $x=0$ is not in the domain of the function $f(x)=log(x)$.
  • Demonstrate that a unique inverse can be found for some functions by restricting the domain

• Introduction to Domain and Range

  • The domain is shown in the left oval in the picture below.
  • The function provides an output value, $f(x)$, for each member of the domain.  
  • In this case, the domain of $f$ is the set of all real numbers except $0$.
  • So the domain of this function is $\mathbb{R}-\{0\}$ .
  • With this knowledge in hand, let's find the domain of a function.

• Domains of Rational and Radical Functions

  • The domain of a rational expression of is the set of all points for which the denominator is not zero.
  • To find the domain of a rational function, set the denominator equal to zero and solve.  
  • All values of $x$ except for those that satisfy $2(x^2-5)=0$ are the domain of the expression.
  • All values of $x$ except for those that satisfy $\sqrt x \geq 0$ are the domain of the function.
  • So, all real number greater than or equal to $3$ is the domain of the function.

• Introduction to Rational Functions

  • The domain is comprised of all values of $x \neq 0$.
  • The domain of this function includes all values of $x$, except where $x^2 - 4 = 0$.
  • The domain of this function is all values of $x$ except those where $x^2 + 2 = 0$.
  • Since this condition cannot be satisfied by a real number, the domain of the function is all real numbers.
  • The domain of this function is all values of $x$ except $+2$ or $-2$.

• Inverse Trigonometric Functions

  • Note that the domain of the inverse function is the range of the original function, and vice versa.
  • We choose a domain for each function that includes the number $0$.
  • Note the domain and range of each function.
  • To find the domain and range of inverse trigonometric functions, we switch the domain and range of the original functions.
  • (a) The sine function shown on a restricted domain of $\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$; (b) The cosine function shown on a restricted domain of $\left[0, \pi\right]$.

• Two-Component Regulatory Systems

  • The N-terminal domain of this protein forms part of the cytoplasmic region of the protein, which may be the sensor domain responsible for sensing turgor pressure.
  • Histidine kinases usually have an N-terminal ligand-binding domain and a C-terminal kinase domain, but other domains may also be present.
  • Members of this family, however, have an integral membrane sensor domain.
  • In addition to the sensor domain and kinase core, they contain a CheY-like receiver domain and a His-containing phosphotransfer (HPt) domain.
  • The blades are formed by two N-terminal domains each, and the compact central hub assembles the C-terminal kinase domains.

• Continuity

  • The function $f$ is continuous at some point $c$ of its domain if the limit of $f(x)$ as $x$ approaches $c$ through the domain of $f$ exists and is equal to $f(c)$.
  • The function $f$ is said to be continuous if it is continuous at every point of its domain.
  • If the point $c$ in the domain of $f$ is not a limit point of the domain, then this condition is vacuously true, since $x$ cannot approach $c$ through values not equal to $c$.

• The Diversity of Life

  • The diversity of life can be classified within the three major domains (Bacteria, Eukarya and Archaea) using phylogenetic trees.
  • The third domain contains the eukaryotes and includes unicellular microorganisms together with the four original kingdoms (excluding bacteria).
  • Woese defined Archaea as a new domain, and this resulted in a new taxonomic tree .
  • Many organisms belonging to the Archaea domain live under extreme conditions and are called extremophiles.
  • The tree shows the separation of living organisms into three domains: Bacteria, Archaea, and Eukarya.

• Regulation by Biosynthetic Enzymes

  • (This differs from eukaryotic cells, where RNA must exit the nucleus before translation starts. ) The attenuator sequence, which is located between the mRNA leader sequence (5' UTR) and trp operon gene sequence, contains four domains, where domain 3 can pair with domain 2 or domain 4.
  • A high level of tryptophan will permit ribosomes to translate the attenuator sequence domains 1 and 2, allowing domains 3 and 4 to form a hairpin structure, which results in termination of transcription of the trp operon.
  • In contrast, a low level of tryptophan means that the ribosome will stall at domain 1, causing the domains 2 and 3 to form a different hairpin structure that does not signal termination of transcription.
  • Thus, domain 4 is an attenuator.
  • Without domain 4, translation can continue regardless of the level of tryptophan.