MLF - Domain and Range (Lesson)

Domain and Range

In the last lesson, we used the linear function $C(m) = 0.15m$ to model the relationship between the data used and the total cost of the cell phone plan. The data used was the independent value or the input and the total cost was the dependent value or the output. Another way to describe the input and the output of a linear function is by determining the domain and the range. 

---

Describing the Domain and Range of a Function

The domain of a function is the set of x-coordinates, or input values, that make sense within the context of the function. The range is the set of y-coordinates, or output values, that make sense for the function. Recall the graph for the function, $C(m) = 0.15m$. 

Notice that the graph of the line did not extend past where x = 0 (at the origin). Why do you think this is? In the context of this problem, it doesn't make sense for the x-value, or input, to be less than 0 because you cannot use a negative amount of data! In the same way, the total cost of the plan could not be negative either. So, the domain for this function would be described as all values greater than or equal to 0 and the range would also be described as all values greater than or equal to 0.

There are two different mathematical formats for describing domain and range as well as describing it verbally. Take a look at the different ways of representing the domain and range. Which format do you prefer?

Verbally	Interval Notation	Set Notation
The domain is the set of all values greater than or equal to 0.	Domain: $[0,\infty)$	$D:{\lbrace x\:\vert\: x\ge0}\rbrace$
The range is the set of all values less than or equal to 0.	Range: $(-\infty,0]$	$R: {\lbrace y\:\vert\: y\le0 }\rbrace$
The domain is the set of all values greater than -3 and less than or equal to 5.	Domain: $(-3, 5]$	$D:{\lbrace x\:\vert\: x>-3\:and\:x\le5}\rbrace$
The range is the set of all real numbers.	Range: $(-\infty,\infty)$	$R:{\lbrace y\:\vert\:y\:ϵ\:}\mathbb{R}\rbrace$

Interval Notation

Let's take a closer look at the interval notation examples in the table above. Sometimes a parenthesis ( ) is used and other times a bracket [ ] is used. Can you compare the interval notation to the verbal description and figure out when each is used? A bracket is used when the domain or range includes the number next to it. For example, in the third row of the table, the domain has to be greater than -3 (cannot equal -3) but CAN equal 5 so the interval notion is $(-3, 5]$, where the -3 is placed next to a parenthesis and the 5 is placed next to a bracket. 

Why do you suppose the infinity symbols (-∞ and ∞) are always placed next to parenthesis? This is because a number cannot actually ever reach infinity or negative infinity since it continues indefinitely! 

Set Notation

Now, let's investigate the set notation examples in the table. There are a few new symbols! The curly braces { } represent the set of numbers. The vertical line in the expression | can be read as "in which" so a whole expression of $D:{\lbrace x\:\vert\: x\ge0}\rbrace$ could be read as "The domain is the set of x-values in which x is greater than or equal to 0". Let's try another. The expression $D:{\lbrace x\:\vert\: x>-3\:and\:x\le5}\rbrace$ can be read as "The domain is the set of x-values in which x is greater than -3 and x is less than or equal to 5".

There are two other new symbols in the last row. The symbol ∈ can be read as "belongs to" and the symbol ℝ  is read as "the set of all real numbers". So, the whole expression of $R:{\lbrace y\:\vert\:y\:ϵ\:}\mathbb{R}\rbrace$ is read as "The range is the set of y-values in which y belongs to the set of all real numbers". It's like learning a new language! Here's a recap of the new symbols you have learned:

Symbol	Explanation
[  ]	Brackets are used to show that a number is included in the set.
(  )	Parenthesis are used to show that a set includes numbers that are either greater or less than the  number shown
{ }	Curly brackets are used to hold all of the numbers in a set.
$\vert\:$	Read as "in which" or "such that"
$ϵ$	Read as "belongs to"
$\mathbb{R}$	Read as "the set of all real numbers"

Example: You find a new cell phone plan that charges you $0.25 per MB of data.

Part I: Create a linear function that represents this relationship.

The linear function for this relationship would be $y=0.25x$ .

Part II: In this plan, you are not allowed to go over 6 MB of data. What are the domain and the range of this function?

The minimum domain and range values would both be 0 since you can't use a negative amount of data or has a negative cost.

Now, let's determine the maximum domain and range values. The maximum y-value is 6MB. We can plug that in to find the maximum x-value.

$6=0.25x$                 Divide by 0.25 to isolate the x.

$24=x$                      The most you can spend is $24. 

Verbal: The domain is the set of all values greater than or equal to zero and less than or equal to 24. The range is the set of all values greater than or equal to 0 and less than or equal to 6.

Interval Notation: Domain: [0, 24] Range: [0, 6] 

Set Notation: $D:{\lbrace x\:\vert\:x\ge0\:and\:x\le24}\rbrace$ $R:{\lbrace y\:\vert\:y\ge0\:and\:y\le6}\rbrace$

---

Domain and Range Practice

Create a linear function that represents the relationship described in each question. Then, determine an appropriate domain and range for the scenario. Write the domain and range verbally, using interval notation and using set notation. 

1. You are paid $10.00 an hour at the skating rink. 

Part I: Create a linear function that represents this relationship.

Part II: You are only allowed to work a maximum of 15 hours a week. What is the domain and the range of this function?

2.  You go shopping for candy. Each piece of candy cost $0.75. 

Part I: Create a linear function that represents this relationship.

Part II: You only have $5.00 on you. What is the domain and the range of this function?

TO VIEW THE SOLUTIONS ONCE YOU HAVE PRACTICED, CLICK HERE. Links to an external site.

IMAGES CREATED BY GAVS