I am professor Allie and I am going to talk about the domain and range of functions. 

Think of domain as all of the possible input values in a function (usually represented by the variable x), this means horizontally across the graph. You can also think of it as the set of all possible input values that correspond to specific output values. When dealing with some functions, like a parabola, there are no defined endpoints, meaning that you do not know exactly where or when the function stops when looking horizontally across the graph. This means that it could stretch as far as it wants from right to left, so the domain would be all x values (-∞, ∞). 

If we were looking at a graph that had defined endpoints, such as this, we would find the domain a little differently. We still look horizontally across the graph, but the line does not go on forever; point A is at -3 and point B is at 2. Since both of the endpoints are filled in, this means that these points are included in the domain, so you would use brackets to show the domain of [-3, 2]. If the endpoints were hollow, this would mean that every number up to but not including the endpoint is in your domain. If this were the case, you would use parentheses instead, so your domain would be (-3, 2).

Here’s an instance where domain becomes slightly more complicated; the equation 

y=3/x-1

Think about properties of division; when you type 8÷0 into your calculator, your calculator probably says “undefined.” This can be transferred over when thinking about domain! If your input or x value is 1, the equation would come out to y=3÷0, which is undefined, meaning that 1 is not part of the domain of this function. This means that your domain would be all real numbers other than x=1, so (-∞,1)∪(1,∞). The “∪” between the two sets of parentheses links them together, giving the valid domain for this equation. 

Range, on the other hand, can be thought of as the set of all possible output values (usually the variable y). Try thinking about range vertically and look at how the function compares to the y axis. Finding and understanding range is very similar to domain; take this simple line going through the origin (0,0). Since there are no defined endpoints and it appears like the line could extend up and down the y axis as long as it wants, this means that the range would be all real numbers (-∞, ∞). 

If there were endpoints on this line, you could follow the endpoint and look at the number that corresponds on the y axis in order to figure out how far the line extends vertically. You also 

follow the same rule as you would for domain when looking at whether or not the endpoints are included and when you should link two sets of parentheses together with a union. 

Let’s look at one more example of range when looking at a parabola. When looking at this graph vertically, you can see that there are no points below zero. This means that the range would be for all real numbers greater than (but including) zero. [0, ∞).

There are two different ways to write domain and range, and these are interval and set builder notation. I have been writing domain and range in interval notation, which is significantly more simple than set builder. Interval notation utilizes parentheses and brackets and all you have to do is plug in the numbers that define the function you are working with. 

Set builder, on the other hand, forces you to consider the types of numbers you are working with; these can range from natural, integers, real and rational numbers to irrational numbers. When writing the domain or range in integer form, you first start with the variable you are writing for (usually x or y). If we were writing the domain of this line, which can be expressed by the equation y=x, we would be writing for the x variable. Then we would need to consider what types of numbers are included in the domain. In this example, all real numbers are included. Then we would show this using the infinity sign. 

x∈R∣-∞<x<∞

The ∈ above indicates that x is an element of real numbers such that x is between negative and positive infinity. 

It is very important that everyone learns and understands how to find the domain and range of a function because it will help you be able to define them. Also, some functions model real world examples and it's important that you are able to recognize that some values are not possible for certain situations.

If you are still confused, I recommend this youtube video to help clarify the points that I have already made.  

https://www.youtube.com/watch?v=fyROLkZc75E