Domain and Range

Part A:

Hi, I will be explaining the concept of domain and range. Domain and range are both actually quite simple, and are good to understand when graphing. Without domain and range, solving a graph algebraically or graphically becomes extremely difficult. Luckily, we won't have to deal with that difficulty after today.

First, it's good to understand conceptually what domain and range values actually are. Let's begin:

- Domain is all the horizontal axis' possible values, or, as traditionally taught, all the "x" values/input values. 
- Range is all the vertical axis' possible values, or traditionally, all the "y" values/output values. 

Just knowing the domain and range isn't enough though, we have to be able to write it in a way that all other mathematicians can understand. The two ways we write domain and range are as follows:

1) Interval Notation: So, if the domain (all possible input/x-values) is equal to all values between (and including) 0 and 5, then the domain in interval notation would be as follows: [0,5]. If the 0 and 5 are not included, then it would be as follows: (0,5). Brackets indicate inclusion of the end point, but parentheses indicate exclusion of the end point. 

2) Set Builder Notation: There are more complicated ways to write domain and range. In set builder notation, the realm of the numbers defined in the domain/range are included. So, with the same interval of 0 and 5 (with end points being excluded), the set builder form of the domain would be as follows: {x ∈ R | 0 < x < 5}. The ∈ is read as "is an element of" and the R is read as "all real numbers," while the | is read as "such that." This gives the condition that while x has the possibility of being a part of all real numbers, that's only true for the condition of all real numbers between and not including 0 and 5. 

That might be a bit confusing, because set builder notation is usually where most people get lost. Here's another example for the more visual learners:

• The  means "a member of" (or simply "in")
• The  is the special symbol for Real Numbers.

Let's do a simple example using both domain and range now: In the graph shown below, the domain is every possible input value, or "x", that we can have on the line segment. So, the domain is anything between -2 and 2. In interval notation, that can be written as [-2,2]. Notice the closed brackets indicating that -2 and 2 are a part of the domain. In set builder notation, the domain is {x ∈ R | -2 ≤ x ≤ 2}. Notice the "≤" which indicates that the numbers outside are included.

The range for the same graph would be all the possible "y" values, or output values, that the line segment can have. In this case, the range would be all values between -4 and 8, including those two end points. In interval notation it would be [-4,8] and in set builder it would be {x ∈ R | -4 ≤ x ≤ 8}.

If you want to see just one more resource, check out these videos on domain and range: 

For domain definition: http://www.virtualnerd.com/algebra-1/relations-functions/domain-definition.php 

For range definition: http://www.virtualnerd.com/algebra-1/relations-functions/range-definition.php

For finding domain and range: http://www.virtualnerd.com/pre-algebra/algebra-tools/find-domain-range-example.php

For interval notation: https://www.youtube.com/watch?v=ow8Ug7_d-9Y

For set builder notation: http://www.virtualnerd.com/tutorials/?id=Alg1_5_1_8

And there you have it! You now officially understand and can communicate the concepts of domain and range for a graph. Nice job!