Domain and Range

For every relation consisting of ordered pairs (x,y), there are two important sets of elements:

The set of all values of the independent variable x is the domain.
The set of all values of the dependent variable y is the range.

Example

Find the domain and range of each relation.

$LaTeX: F=\lbrace\left(3,4\right),\left(-4,1\right),\left(2,7\right)\rbrace$ $F=\lbrace\left(3,4\right),\left(-4,1\right),\left(2,7\right)\rbrace$

$LaTeX: G=\lbrace\left(3,2\right),\left(3,3\right),\left(3,7\right),\left(4,5\right)\rbrace$ $G=\lbrace\left(3,2\right),\left(3,3\right),\left(3,7\right),\left(4,5\right)\rbrace$

$LaTeX: H=\lbrace\left(2,2\right),\left(3,2\right),\left(4,2\right),\left(5,-1\right)\rbrace$ $H=\lbrace\left(2,2\right),\left(3,2\right),\left(4,2\right),\left(5,-1\right)\rbrace$

Solution

The domain is the set of x-values. So, for F the domain is $LaTeX: \lbrace3,-4,2\rbrace$ $\lbrace3,-4,2\rbrace$ , for G the domain is $LaTeX: \lbrace3,4\rbrace$ $\lbrace3,4\rbrace$ , and for H the domain is $LaTeX: \lbrace2,3,4,5\rbrace.\:$ $\lbrace2,3,4,5\rbrace.\:$

The range is the set of y-values. So, for F the range is $LaTeX: \lbrace4,1,7\rbrace$ $\lbrace4,1,7\rbrace$ , for G the range is $LaTeX: \lbrace2,3,5,7\rbrace$ $\lbrace2,3,5,7\rbrace$ , and for H the domain is $LaTeX: \lbrace2,-1\rbrace.$ $\lbrace2,-1\rbrace.$

Agreement on Domain

Unless specified otherwise, the domain of a relation is assumed to be all real numbers that produce real numbers when substituted for the independent variable.

For example, the range of the function LaTeX: y=4x-5 $y=4x-5$ is the set of all real numbers, $LaTeX: \mathbb{R}.$ $\mathbb{R}.$ This is because any real number can be used as a replacement for x in this equation.

Finding Domain and Ranges from Graphs

Take a look at this video about how to get the domain and range from a graph, plus a little review of interval notation.