There are multiple ways to write the Domain or Range of a Function. First, I will provide quick definitions of Domain and Range and then, with the help of an Example, I will explain the common mistakes that most of the Students make. So, Here are the definitions:
- Domain: The set of all possible input values ("x" values) for which the function is defined.
- Range: The set of all possible output values ("y" values) generated by the function.
Correctly identifying these sets is necessary for solving problems and understand the behavior of Functions. Now, let's look at the common errors students make.
Example:
We know that if x is in the interval - 3 < x < 3, a square root of a negative number is obtained. Hence, no Real Number for our given f (x) exists when we take x from that interval. Therefore, there are multiple ways to write the Domain and Range of this Function.
For Domain:
Also, you can write it as;
OR;
ℝ - (-3, 3) or ℝ \ (-3, 3)
All three Notations are correct and valid.
Mistakes:
Invalid Notation, Always use a Round Bracket (Parentheses) 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.
If you want to Learn more about Brackets and their Correct use Click Here! I have already explained about the correct usage of all the three types of Brackets in that Article, along with Examples and their complete details.
This is also incorrect notation because it excludes the endpoints x = - 3 and x = 3, which are valid values for x. So, whenever you know that the End-Points are Included, always use Square Brackets.
ℝ - {-3, 3} or ℝ \ {-3, 3}
And If you write it like that, this will also be the incorrect notation. Because, 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. So, if you want to write the Domain like this, the correct way is to use the Round Brackets so that you can subtract that specific interval.
For Range:
Range f = Set of all non-negative Real numbers
OR,
Both the Notations are correct and valid.
Mistakes:
Range f = Set of all positive Real numbers
This is incorrect because it excludes 0, even though 0 is also a valid output of our function. The Set of Positive Real numbers contains only positive numbers that does not include 0, as 0 is a Neutral Entity. Similarly, if the Domain or Range is described as the Set of all Negative Real numbers, it means that it does not include positive numbers as well as 0.
Conclusion:
Understanding and correctly writing and determining the Domain and Range of a Function is necessary for solving Mathematical Problems and analyzing Functions accurately. Always pay attention to proper notation, including the use of Brackets and ensure End-Points and specific values are correctly included or excluded as needed. Avoid common mistakes such as misusing infinity Brackets, excluding valid values like 0, or incorrectly using curly braces for intervals. By following these guidelines, Students can confidently determine the Domain and Range of any function.
0 Comments