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Understanding the Domain and Range of the Function f(x) x-1
Understanding the Domain and Range of the Function f(x) x-1
Understanding the domain and range of a mathematical function is essential for grasping its behavior and applications. In this article, we will explore the function f(x) x-1 and dissect its domain and range. We'll cover both analytical methods and graphical representations to enhance your understanding.
Introduction
Functional analysis often involves identifying and defining the domain and range of a given function. This helps in comprehending the set of all possible inputs and the corresponding outputs the function can produce. Let's begin by defining these crucial terms:
Domain: The set of all possible input values for which the function is defined. Range: The set of all possible output values that the function can produce.Understanding the Function f(x) x-1
The function in question here is f(x) x-1. This is a linear function that subtracts 1 from the input value x. Let's explore how we determine its domain and range.
Domain of f(x) x-1
For the function f(x) x-1, the domain is the set of all real numbers, denoted as R. This is because f(x) can take any real number input and produce a real number output. In mathematical notation, the domain is:
Domain: R (real numbers)
To illustrate, if we take an arbitrary value for x, say x -2, then:
fx x - 1 -2 - 1 -3
Thus, the function can handle any real number input.
Range of f(x) x-1
Understanding the range of f(x) x-1 involves considering all possible outputs. Let's break it down step-by-step:
We start by considering the behavior of f(x) x-1 as x varies. For any real number x, x-1 will also be a real number, and it can cover all values from negative infinity to positive infinity. Therefore, the smallest value f(x) can achieve is 0 (when x1), and it can increase indefinitely.Mathematically, the range can be expressed as:
Range: [0, ∞)
This means the function starts from 0 and extends infinitely in the positive direction.
Graphical Representation
Graphical analysis provides a visual understanding of the function. The graph of f(x) x-1 is a straight line with a slope of 1 and y-intercept -1. From the graph, you can see:
The domain is all real numbers, as the line extends infinitely in both directions horizontally. The range starts at 0 and goes to positive infinity.The graph helps visualize the boundedness of the range and the unbounded nature of the domain.
Conclusion
In summary, the function f(x) x-1 has a domain that includes all real numbers, denoted by R. Its range starts from 0 and extends indefinitely to positive infinity, represented as [0, ∞). Both analytical and graphical approaches support these conclusions, providing a comprehensive understanding of the function's behavior.