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 $\left[0, \pi\right]$.

• One-to-One Functions

  • A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its codomain.
  • A one-to-one function, also called an injective function, never maps distinct elements of its domain to the same element of its co-domain.
  • In other words, every element of the function's range corresponds to exactly one element of its domain.
  • Another way to determine if the function is one-to-one is to make a table of values and check to see if every element of the range corresponds to exactly one element of the domain.  

• Piecewise Functions

  • Example 2: Graph the function and determine its domain and range:
  • For the last part (piece), $f(x)=x$ for the domain $x>2$, a few ordered pairs are:
  • This has to do with the specific domains for each part of the function.  
  • Each part of the function is graphed based upon the specific domain chosen.  
  • Depending on the value of the domain, each piece is different.

• Relative Minima and Maxima

  • In mathematics, the maximum and minimum of a function (known collectively as extrema) are the largest and smallest value that a function takes at a point either within a given neighborhood (local or relative extremum) or within the function domain in its entirety (global or absolute extremum).
  • While some functions are increasing (or decreasing) over their entire domain, many others are not.  
  • Note that we have to speak of local extrema, because any given local extremum as defined here is not necessarily the highest maximum or lowest minimum in the function’s entire domain.
  • The graph attains an absolute maximum in two locations, $x=-2$ and $x=2$, because at these locations, the graph attains its highest point on the domain of the function.
  • The graph attains an absolute minimum at $x=3$, because it is the lowest point on the domain of the function’s graph.

• Inverses of Composite Functions

  • If $f$ is an invertible function with domain $X$ and range $Y$, then
  • This statement is equivalent to the first of the above-given definitions of the inverse, and it becomes equivalent to the second definition if $Y$ coincides with the co-domain of $f$.