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## Algebra I (2018 edition)

### Course: Algebra I (2018 edition)>Unit 6

Lesson 5: Introduction to the domain and range of a function

# Intervals and interval notation

Introducing intervals, which are bounded sets of numbers and are very useful when describing domain and range.

We can use interval notation to show that a value falls between two endpoints. For example, -3≤x≤2, [-3,2], and {x∈ℝ|-3≤x≤2} all mean that x is between -3 and 2 and could be either endpoint.

## Want to join the conversation?

• what kind of R is that, is it a maths symbol or that's just your way of writing R •   It's a mathematical symbol, ℝ, meaning "the real numbers".
You may also see, from time to time:
ℕ - the natural numbers
ℤ - the integers
ℚ - the rational numbers (quotients)
ℂ - the complex numbers.
• Instead of writing x<1 or x>1 can I write x<1 U x>1 • You sure can, as x<1 or "x>1" basically means "x<1 U x>1".
Just to make it clear, U is ( as most people who use sets would know ) union. And the union between, suppose A and B ( where A and B are set) which would be written as A U B would mean values that belong to set A or set B. You can also think of it as values/objects that are part of the whole of set A and set B ( A + B ).

Hope that helped. :} :] :)
• Can you please calrify for me what exactly does "real numbers"mean. •  The real numbers are the set of numbers including rational and irrational numbers. So numbers like 6/7, 0.1, 3000, pi, etc. are included. However, a number like "i" is not included. "i" is a complex number. It is equal to the square root of -1. One way to define real numbers is a number that can be plotted on the number line like the one Sal was using in this video. :)
• Usage of '(' and ')' after and before the infinity denotes its included. However, how can we handle infinity which is just an idea. So is it better to use '[' and ']' brackets before and after the infinity? • At , Sal said that open circles indicate an open interval. Could't he just make it a closed interval by making it {-0.99999,3.99999} instead of {-1,4}? • No, but cause values like -0.9999999 and 3.99999999 are also in the solution set.
The solution set must include all possible solutions. This is why open intervals are used. They indicate that we want to start at -1, but not include it and we want all numbers up to but not including the 4.

Hope this helps.
• This is probably a stupid question but around he closes a parentheses with a bracket, can you do that? I always thought you had to close your bracket or parentheses with another bracket or parenthesis but you couldn't interchange them like that. • This is actually a pretty good question and it's good to see that you ate paying attention.

In Interval Notation, you actually can have a parenthesis on one side and a bracket on the other and have the notation be correct. Keep in mind that this is a type of mathematical notation and not Grammar of any language that uses this type of punctuation.

If you go back and watch Sal show the notation of the first interval (blue writing), both boundary expressions include "equal to" (`x` is greater than or equal to -3 and `x` is less than or equal to 2). When the boundary includes the number (equal to), we use the square brackets to notate the interval.

When Sal shows the notation for the second interval (pink writing), the boundary expressions do NOT include "equal to" (`x` is greater than -1 and `x` is less than 4). When the boundary does NOT includes the number (greater than or less than, but not equal to), we use parentheses to notate the interval.

In the third example (the one you are citing), the one boundary expression includes "equal to" and one does not (`x` is greater than -4 and `x` is less than or equal to -1). In this case we mix the parenthesis and the square bracket. We have a parenthesis in front of -4 because -4 is NOT included in the interval. We have a square bracket after -1 because -1 IS included in the interval.

If the boundary expressions do both things, we use both notations.
• At ,Shouldn't it be {x<1 and x>1} instead of {x<1 or x>1}? • Is there a name for the brackets that look like wavy lines? What do they signify? Thanks. • What is a real number? • A real number is really just any number that can be defined in an equation and a graph, whether it's rational or irrational (if you're familiar with those terms, you would be able to see where I'm coming from).

Some examples that are real numbers:
8, 1/45, pi symbol (3.1415926...)

Examples of an imaginary numbers (the opposite of a real number):
8/0, square root of -5 (these numbers would be undefined as they aren't what people would see as practically real. If those people were talking mathematically however, which isn't something people would do at a regular basis, then they wouldn't perceive imaginary numbers as something we can put immediately in an equation or graph).

Hopefully that's straightforward enough for you to understand. 