What is the Correct Way to use Brackets while writing the Domain and Range of a Function?

Before knowing the correct use of Brackets, you should know about the types of the Brackets. So, I will explain about each type of  Bracket and the correct way to use them, one by one.

1. Round Brackets (Parentheses) - ( )

Round Brackets are the most commonly used Brackets in Mathematics. They are mainly used to denote that an endpoint is not included in the Domain and Range. In other words, if you see a Round Bracket around a value, it means that this value is not included in the interval or Set of Domain and Range. 

Examples:

1. (a, b): This means that the Domain or Range includes values greater than a and less than b, but not a and b themselves.
2. (0, 10): This shows that the Function can take any value greater than 0 and less than 10, but 0 and 10 are not included.   

Important to Note:

        (-∞, +∞): Always use a round bracket with negative and positive infinity. Infinity is not a number, so it can never be included in an interval. It is just a way or a sign to denote something undefined.

        (-5, +∞): Whenever you get an Infinity in your Domain or Range it means that the values go up to infinity, as in this case where the Domain or Range starts from -5 and tend towards positive infinity that include all the positive values. We don't know where its gonna end, so that's why we use the +∞ to show that.

        ]-5, +∞[ : At some places, you might see Brackets like this, maybe in the Higher Classes. These are also use when the interval is open and are called the Open Brackets, though they are more commonly known as open interval notation. While "Open Brackets" is not a universally standard term, it can refer to this notation style in specific contexts or regions. You might also hear people refer to open brackets to indicate that the interval is open, whether it is open from one side or both. These are Square Brackets, about which I am going to explain next, but they are used differently in this context.

2. Square Brackets - [ ]

Square Brackets are used when an endpoint is included in the Domain or Range. This means that the value represented by the square bracket is a part of the set. 

Examples:

1. [a, b]: This means that the values include a and b, as well as all the values in between.
2. [-3, 10]: This shows that the Function can take any value between -3 and 10, including both -3 and 10.

Important to Note:

Sometimes, we have a Domain or Range like this;

        (3, 10]: This interval represents all values between 3 and 10, where 3 is not included, but 10 is included. Sometimes, we use both types of brackets at the same time, as in this example. The Round Bracket on 3 indicates that 3 in not a part of the interval, means that the values start just after 3. The square bracket on 10 shows that 10 is included, meaning the interval goes up to and included 10. 

        (-∞, 2]: This interval represents all values less than or equal to 2, with 2 included in the interval and extending infinitely in the negative direction. Again, I want you to remember to always use Round Brackets with the infinity sign, whether positive or negative Infinity.

        This type of mixed interval is often used when one endpoint is fixed and included, while the other is excluded, providing flexibility in defining Ranges or Domains in mathematical contexts.

3. Curly Braces - { }

Curly braces are used when you directly listing specific elements or values in a set, typically for discrete values, not continuous intervals. These are often used to represent a finite set of values. 

Discrete values refer to distinct, separate values that can be counted or listed individually. These values are not continuous and do not have intermediate values between them.

Examples:

1. {1, 2, 3} : This means that only the values 1, 2, and 3 are the part of the Domain or Range.

Important to Note:   

         ℝ - {2} or ℝ \ {2} :  Sometimes, we encounter Functions whose Domain or Range includes the entire Set of Real Numbers, with just one value or element excluded. In this case, we write it like this and use curly braces for that specific value to indicate that it is being removed from the whole set. Both notations are correct.

Conclusion:   

         Brackets play a vital part in Mathematical Notations, especially when dealing with Functions and their Domains and Ranges. Understanding how to use Round, Square and Curly Brackets properly will help you represent mathematical Sets accurately and communicate your Ideas clearly.