Domain and Range
Domain refers to all the inputs (or \(x\)) values that are possible for a given function. It is also referred to as the replacement set. and is represented algebraically as \(\mathbf{D}\). Range refers to the set of outputs (or \(y\)) values that are possible for a given function. It is also referred to as the solution set and is represented algebraically as \(\mathbf{R}\).
A Set is a collection of numbers or other symbols.
Set Notation
Set Notation refers to the way that sets are represented and relate to each other. It incorporates various symbols including:
- \(\in\) represents a member/element of
- \(\notin\) represents NOT a member/element of
- \(\mathbb{R}\) represents Real Numbers
- \(\mathbb{N}\) represents Positive Integers
- \(\mathbb{W}\) represents Postive Integers and 0
- \(\mathbb{Z}\) represents All Integers and 0
- \(ℚ\) represents Rational Numbers
- \(\mathbb{I}\) represents Irrational Numbers
- \(\{ \}\) represents a set/collection of elements
- \(|\) represents for which/so that
- \(:\) represents for which/so that
Common Number Sets
Real Numbers
A Real Number Set is defined as any set of numbers that includes integers, decimals, and/or fractions.
An example of a Real Number Set is \(\{x \in \mathbb{R} | -10 < x < 3\}\).
Natural Numbers
A Natural Number Set is defined as any set of integers that are above \(0\).
An example of a Natural Number Set is \(\{x \in \mathbb{N} | 1,2,3\}\).
Whole Numbers
An Whole Number Set is defined as any set of positive integers including \(0\).
An example of an Integer Set is \(\{x \in \mathbb{W} | 0,3,5\}\).
Integers
An Integer Set is defined as any set of whole numbers, either positive or negative.
An example of an Integer Set is \(\{x \in \mathbb{Z} | -4,0,9\}\).
Rational Numbers
An Rational Number Set is defined as any set of numbers, either positive or negative, that can include integers, decimals, or fractions.
An example of an Integer Set is \(\left\{x \in \mathbb{Q}| -0.5, 6, \cfrac{7}{2}\right\}\).
Complex Numbers
An Irrational Number Set is defined as any set of Integer or Rational Numbers, including rationals that can't be represented as simple fractions.
An example of an Integer Set is \(\left\{x \in \mathbb{I} | 8, -3, \cfrac{2}{\sqrt{8}}\right\}\).
Restrictions
When finding the Domain and Range of a function, only Real Numbers can be included in either set of values.
Domains may contain restricitons for either of the following reasons:
- Dividing by \(0\)
- Taking the square (or even) root of a negative number
In either instance, this would result in an ERROR message on your calculator!!
Dividing by 0
This restriction only occurs with Rational Functions when the denominator is set to \(0\). This restriction only involves one value. Let's use the following function:
This function would provide us with an actual value UNLESS we substituted \(x\) with \(4\):
\(= \cfrac{1}{4-4}\)
\(= \cfrac{1}{0}\)
\(= \text{ERROR!!}\)
When \(x=4\), we will get an ERROR message! Therefore, the restriction in this instance is \(\boldsymbol{x \ne 4}\).
Square Root of Negative Number
This restriction only occurs with Radical Functions and can occur with any even root (ie 4th, 6th, 8th root). This restriction normally involves a range of values. Let's use the following function:
This function would provide us with an actual value UNLESS we substituted \(x\) with any value lower than \(-6\):
\(= \sqrt{-10+6}\)
\(= \sqrt{-4}\)
\(= \text{ERROR!!}\)
When \(x=-10\), we will get an ERROR message! Therefore, the restriction in this instance is \(\boldsymbol{x \lt -6}\).
Example
1. Find the Domain and Range of the set {\((2,5), (3,5), (4,10), (5,0)\)}.
\(\mathbf{D} = \boldsymbol{\{x\in\mathbb{N} | 2,3,4,5\}}\)\(\mathbf{R} = \boldsymbol{\{y\in\mathbb{W} | 5,10,0\}}\)
In this case, we can identify the Domain as the set of \(x\)-values and the Range as the set of \(y\)-values within the set.
2. Find the Domain and Range of the following graph:
\(\mathbf{D} = \boldsymbol{\{x\in\mathbb{R}\}}\)
\(\mathbf{R} = \boldsymbol{\{y\in\mathbb{R} | y \ge 0\}}\)
In this case, we can identify the Domain as a set of real numbers (ie all numbers, positive, negative, and \(0\)). We can identify the Range as a set of real numbers that's equal or greater than \(0\).
From this mapping diagram, we can identify both the Domain and Range:
\(\mathbf{D} = \boldsymbol{\{x\in\mathbb{Z} | 5,-8,19\}}\)\(\mathbf{R} = \boldsymbol{\{x\in\mathbb{N} | 84,7,21\}}\)
In this case, we can identify the Domain as the set of \(x\)-values and the Range as the set of \(y\)-values within the set.
From this graph, we can identify both the Domain and Range:
\(\mathbf{D} = \boldsymbol{\{x\in\mathbb{R} | -5 \le x \le 5\}}\)\(\mathbf{R} = \boldsymbol{\{y\in\mathbb{R} | -5 \le y \le 5\}}\)
In this instance, both the Domain and Range are elements of real numbers equal or greater than \(-5\) and equal or less than \(5\).
\(f(x) = 5x + 3\)
We can begin by drawing a graph of the given function:
From this graph, we can identify both the Domain and Range:
\(\mathbf{D} = \boldsymbol{\{x\in\mathbb{R}\}}\)\(\mathbf{R} = \boldsymbol{\{y\in\mathbb{R}\}}\)
In this instance, both the Domain and Range are elements of real numbers (both encompass all possible numbers).