Find The Domain And Range Of The Function F(x) = (4-x)/(x-4).
In the realm of mathematics, understanding functions is crucial, and two fundamental aspects of any function are its domain and range. The domain specifies the set of all possible input values (often denoted as 'x') for which the function is defined, while the range represents the set of all possible output values (often denoted as 'f(x)' or 'y') that the function can produce. Determining the domain and range is essential for comprehending the behavior and characteristics of a function. In this article, we will delve into the process of finding the domain and range of a specific real-valued function, f(x) = (4-x)/(x-4), providing a step-by-step explanation and illustrative examples. This detailed analysis will not only clarify the concepts of domain and range but also equip you with the skills to analyze similar functions effectively. We'll explore why certain values are excluded from the domain, how the function behaves near these exclusions, and how this behavior impacts the overall range of the function. By the end of this discussion, you will have a solid grasp of how to identify and express the domain and range for rational functions, enhancing your mathematical problem-solving capabilities. This exploration is a cornerstone of understanding more advanced concepts in calculus and analysis, making it a vital skill for anyone pursuing further studies in mathematics or related fields. So, let’s embark on this journey to unravel the intricacies of domain and range, using f(x) = (4-x)/(x-4) as our guide.
Defining the Function: f(x) = (4-x)/(x-4)
The function we are examining is f(x) = (4-x)/(x-4). This is a rational function, a type of function that is expressed as the quotient of two polynomials. In this case, both the numerator (4-x) and the denominator (x-4) are linear polynomials. Rational functions are ubiquitous in mathematics and have wide-ranging applications in various fields, including physics, engineering, and economics. However, they come with their own set of considerations, especially when determining their domain and range. The key characteristic of rational functions that impacts their domain is the presence of a denominator. A fundamental rule in mathematics is that division by zero is undefined. Therefore, any value of x that makes the denominator equal to zero must be excluded from the domain of the function. In the case of f(x) = (4-x)/(x-4), we need to identify any values of x that would cause x-4 to equal zero. Understanding this restriction is paramount to accurately defining the domain of the function. This initial step sets the stage for a comprehensive analysis, ensuring that we avoid mathematical inconsistencies and accurately interpret the function's behavior. By recognizing the potential for division by zero, we can begin to appreciate the nuances of rational functions and their unique properties. This careful consideration of the function's form is the first step toward a complete understanding of its behavior and its place within the broader landscape of mathematical functions. We will further analyze this function to simplify it and then delve deeper into finding its domain and range.
Determining the Domain
The domain of a function is the set of all possible input values (x-values) for which the function is defined. For the given function, f(x) = (4-x)/(x-4), we need to identify any values of x that would make the function undefined. As we discussed earlier, rational functions are undefined when the denominator is equal to zero. Therefore, to find the domain, we need to determine the values of x for which the denominator, x-4, equals zero. Setting x-4 = 0, we can easily solve for x: x = 4. This tells us that x = 4 is the value that makes the denominator zero, and consequently, the function undefined. Therefore, x = 4 must be excluded from the domain. The domain of f(x) is all real numbers except for 4. We can express this in several ways. In set notation, the domain is {x ∈ ℝ | x ≠ 4}, which reads as “the set of all x belonging to the set of real numbers such that x is not equal to 4.” In interval notation, we can represent the domain as (-∞, 4) ∪ (4, ∞). This notation indicates all real numbers less than 4 and all real numbers greater than 4, effectively excluding 4 itself. Understanding how to express the domain in these different notations is crucial for clear and concise communication in mathematics. Now that we have established the domain, we can proceed to analyze the range of the function. The domain provides a foundation for understanding the possible outputs of the function, and by excluding x = 4, we avoid the undefined behavior that would otherwise complicate our analysis. Accurately determining the domain is a critical first step in understanding the full scope of a function's behavior.
Simplifying the Function
Before we determine the range of the function, it's beneficial to simplify f(x) = (4-x)/(x-4). Simplification can often reveal hidden properties of a function and make the process of finding the range more straightforward. Notice that the numerator and denominator are very similar, but they have opposite signs. We can factor out a -1 from the numerator: 4 - x = -(x - 4). Now, we can rewrite the function as f(x) = -(x - 4) / (x - 4). As long as x ≠ 4 (which we already established in the domain), we can cancel the (x - 4) terms in the numerator and denominator. This simplification yields f(x) = -1, for x ≠ 4. This simplified form is much easier to work with. It tells us that for any value of x in the domain (i.e., any x not equal to 4), the function will output -1. However, it is crucial to remember the condition x ≠ 4. The original function was undefined at x = 4, and even though the simplified form doesn't explicitly show this, the restriction still applies. The simplification helps us understand the constant nature of the function over its domain, but it's vital to maintain awareness of the original function's limitations. This process of simplification is a powerful tool in mathematical analysis, but it must be applied with careful attention to the original conditions and restrictions. By recognizing and preserving the exclusion at x = 4, we ensure that our analysis remains accurate and consistent with the initial function definition. With this simplification, determining the range becomes significantly easier, as we have reduced the function to a constant value (almost everywhere).
Finding the Range
Now that we have simplified the function to f(x) = -1 (for x ≠ 4), determining the range becomes a much simpler task. The range of a function is the set of all possible output values (y-values or f(x)-values) that the function can produce. In this case, the simplified function tells us that for any valid input x (any x not equal to 4), the function will always output -1. This might lead one to initially think that the range is simply the set containing only -1, which can be written as {-1}. However, it's important to recall why we excluded x = 4 from the domain. The original function, f(x) = (4-x)/(x-4), is undefined at x = 4 due to division by zero. This means there is no corresponding output value for x = 4. Even though the simplified function f(x) = -1 suggests a constant output, we must consider the original function's behavior. Since x = 4 is not in the domain, the function never actually takes the value -1 at x = 4; it's simply not defined there. Therefore, the range of the function is indeed {-1}. The function only ever outputs the single value -1. This example highlights the importance of considering both the simplified form and the original form of a function when determining its range. The simplification made the constant output clear, but the domain restriction from the original function confirmed that there are no other possible output values. This careful analysis ensures an accurate and complete understanding of the function's behavior. Thus, we can confidently state that the range of f(x) = (4-x)/(x-4) is the singleton set {-1}.
Expressing the Domain and Range
To summarize our findings, let's clearly express the domain and range of the function f(x) = (4-x)/(x-4). We determined that the domain is all real numbers except for x = 4. This exclusion is necessary because x = 4 makes the denominator of the original function equal to zero, resulting in an undefined expression. We can express the domain in various notations: - In set notation: x ∈ ℝ | x ≠ 4} - In interval notation This notation clearly indicates that the only possible output of the function is -1. Understanding how to express the domain and range using both set notation and interval notation is crucial for effective mathematical communication. These notations provide a concise and unambiguous way to describe the sets of input and output values for a function. By clearly stating the domain and range, we provide a complete picture of the function's behavior, highlighting its limitations and its potential outputs. This comprehensive analysis ensures that anyone working with the function understands its properties and can use it appropriately in further mathematical contexts. The combination of a clear domain and range definition forms the foundation for more advanced analysis and applications of the function.
Graphical Representation
A graphical representation can often provide a visual understanding of the domain and range of a function. The function f(x) = (4-x)/(x-4), which simplifies to f(x) = -1 for x ≠ 4, has a straightforward graphical representation. The graph is a horizontal line at y = -1, but with a crucial distinction: there is a hole or a discontinuity at x = 4. This hole represents the fact that x = 4 is not in the domain, and therefore, there is no corresponding y-value at that point. The horizontal line at y = -1 visually confirms our determination of the range as {-1}. The function only ever outputs -1, except at x = 4 where it is undefined. The hole at x = 4 is a critical feature of the graph. It illustrates the importance of considering domain restrictions when interpreting graphical representations of functions. While the simplified equation f(x) = -1 might suggest a continuous horizontal line, the original function's domain restriction necessitates the hole. To accurately graph this function, one would draw a horizontal line at y = -1 and then place an open circle at the point (4, -1) to indicate the discontinuity. This graphical representation not only reinforces our analytical findings but also provides an intuitive understanding of the function's behavior. Visualizing the graph can help in quickly identifying the domain and range, especially for more complex functions. The combination of analytical methods and graphical representations is a powerful approach to fully understanding the properties of functions. In this case, the graph clearly illustrates the domain exclusion and the single value in the range, solidifying our comprehension of f(x) = (4-x)/(x-4).
Conclusion
In conclusion, we have thoroughly analyzed the function f(x) = (4-x)/(x-4) to determine its domain and range. We identified that the domain is all real numbers except for x = 4, expressed as {x ∈ ℝ | x ≠ 4} or (-∞, 4) ∪ (4, ∞). This exclusion arises from the fact that the denominator of the function, x-4, becomes zero at x = 4, leading to an undefined expression. Simplifying the function to f(x) = -1 (for x ≠ 4) allowed us to easily identify the range. The range consists of the single value -1, represented as {-1}. Despite the simplification, we emphasized the importance of remembering the original function's domain restriction, ensuring an accurate determination of the range. We also discussed the graphical representation of the function, which is a horizontal line at y = -1 with a hole at x = 4. This visual aid reinforced our analytical findings, providing an intuitive understanding of the function's behavior. This exercise demonstrates the crucial steps involved in finding the domain and range of a function, particularly for rational functions. Identifying domain restrictions, simplifying the function when possible, and carefully considering the original function's behavior are all essential components of the process. By mastering these techniques, one can confidently analyze a wide range of functions and understand their properties. This understanding is fundamental to further studies in mathematics and its applications. The ability to determine the domain and range is not just a mathematical exercise; it is a core skill that underpins many areas of science, engineering, and beyond. Through this detailed exploration of f(x) = (4-x)/(x-4), we have hopefully provided a clear and comprehensive guide to this essential concept.
Find the domain and range of the real valued function f(x) given by f(x) = (4-x)/(x-4).