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Course: Get ready for Precalculus>Unit 3

Lesson 5: Intro to 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.

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• 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?
• No... you have the symbols reversed. The square brackets indicate the numbers are in the set.
For example: [5, infinity) is the same as x >= 5.
Hope this helps.
• 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.
• At ,Shouldn't it be {x<1 and x>1} instead of {x<1 or x>1}?
• No... the video if correct. Using the word "and" means that X would need to be both less than 1 and greater than 1 at the same time. This is impossible. The word "or" means that you just need one of the conditions to be true.
Hope this helps.
• Is there a name for the brackets that look like wavy lines? What do they signify? Thanks.
• I just call them curly brackets. You may see them used in place of regular parentheses in equations and expressions. The one time they must be used is when a set is in roster form of set notation.
Roster form for set or natural numbers: {1, 2, 3, 4 ...}
Set notation: {x | x ϵ Integers and x > 0}

Hope this helps.
• 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.