Domain and Range


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Definition of Domain and Range

 

 

Example 1

The mapping diagram lists all ordered pairs belonging to a function \(g\).

The mapping diagram shows 0 mapping to 9, 1 mapping to 4, 2 mapping to 1, 3 mapping to 0, 4 mapping to 1, 5 mapping to 4, and 6 mapping to 9.

 

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Example 2

Determine the domain and range of \(y=f(x)\).

A function consists of the points (negative 4, negative 3), (negative 2, 0), (negative 1, 1), (2, 3), and (3, negative 3).

Solution

The domain of a function is the set of all valid inputs.

  • In other words, the domain is the set of all \(x\)-values of the function.
  • This graph exists at these \(x\)-values: \(-4\), \(-2\), \(-1\), \(2\), and \(3\).

Therefore, the domain is \(D=\{-4,~{-2},~{-1},~2,~3\}\).

The range of a function is the set of all possible output values.

  • In other words, the range is the set of all \(y\)-values of the function.
  • This graph has points with \(y\)-values \(-3\), \(0\), \(1\), and \(3\).

Therefore, the range is \(R=\{-3,~0,~1,~3\}\).

Note that, while the \(y\)-value of \(-3\) appears twice on the graph, we list it only once when writing the range.

Further, we normally list the elements in increasing order, though that is not a requirement when listing the elements of a set.


Check Your Understanding 1


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Revisiting Several Definitions

Recall

A relation is a set of ordered pairs.

Previously, you may have encountered definitions of function, one-to-one, and many-to-one. These terms can be defined using the terms domain and range.

A function is a relation in which each element of the domain corresponds to exactly one element of the range.

A function is one-to-one if each element of the range corresponds to exactly one element of the domain.

A function is many-to-one if there exists at least one element in the range that corresponds to different elements in the domain.

For example, \(f(x)=x^2\) is a many-to-one function but \(f(x)=x^3\) is a one-to-one function:

The quadratic function y equals x squared has two x values, 2 and negative 2, which result in a y value of 4.

\(f(x)=x^2\) is many-to-one since there exists an element of the range that corresponds to different elements in the domain. For instance, \(y=4\) corresponds to two different input values, namely, \(x=2\) and \(x=-2\).

The cubic function y equals x to the exponent 3 has a unique x value for every value of y.

\(f(x)=x^3\) is one-to-one since each element of the range corresponds to exactly one element of the domain. There are no \(y\)-values that correspond to more than one \(x\)-value.