Understanding the Domain and Range of the Function f(x) x - x: A Comprehensive Guide
In mathematics, particularly within the context of functions and relations, the concepts of domain and range are fundamental. These terms describe the sets of possible input and output values, respectively. Let's explore the domain and range of the function f(x) x - x.
Introduction to Domain and Range
For a function f: A → B, the domain is the set of all possible input values (x-values) for which the function is defined, and the range is the set of all resulting output values or y-values. In other words, the domain of a relation is the set of all possible x-values, and the range is the set of corresponding y-values for those x-values.
Exploring the Function f(x) x - x
Given the function f(x) x - x, let's break down its domain and range.
The Domain
The domain of the function is the set of all real numbers, R. This is because the function is defined for all real numbers x, and there are no restrictions on the values that x can take. The expression x - x is always 0 for any real number x. Therefore, the domain of f(x) is R or the set of all real numbers.
The Range
Since the function f(x) x - x simplifies to 0 for every input x, the range of the function is also a single value, 0. In set notation, the range can be represented as the set containing only the number 0, which is {0}.
Representing the Graph of the Function
The graph of the function f(x) x - x is a horizontal line at y 0 for all values of x. The domain of the function is the x-axis, which is the set of all real numbers, and the range is the y-axis, which contains only the value 0.
Conclusion
In summary, the domain of the function f(x) x - x is the set of all real numbers R, and the range is the set {0}. This function, although seemingly complex, simplifies to a constant function, which always outputs 0, regardless of the input.
Further Reading
For a deeper understanding of functions, domain, and range, consider exploring additional concepts such as real numbers, modulus functions, and the domain and range of more complex functions.