Domain Of Absolute Value Function F(x) = 2|x-1| + 3

In mathematics, the domain of a function is a fundamental concept that defines the set of all possible input values (often denoted as x) for which the function produces a valid output. In simpler terms, it's the range of values you can plug into a function without encountering any mathematical impossibilities, such as division by zero or taking the square root of a negative number. Identifying the domain is crucial for understanding the behavior and limitations of a function. This article delves into the process of determining the domain of the function f(x) = 2|x-1| + 3, providing a comprehensive explanation and step-by-step analysis. We'll explore the properties of absolute value functions and how they influence the domain, ensuring you grasp the underlying principles and can apply them to similar problems. Understanding the domain isn't just about finding the right answer; it's about developing a deeper understanding of how functions work and the relationships between inputs and outputs. The ability to determine the domain of a function is a critical skill in various areas of mathematics, including calculus, algebra, and analysis. It allows us to define the boundaries within which a function operates meaningfully, ensuring that we are working with valid and consistent results. This article aims to equip you with the knowledge and tools necessary to confidently tackle domain-related problems, empowering you to excel in your mathematical endeavors. We'll break down the concept into manageable parts, making it accessible and engaging for learners of all levels. So, let's embark on this journey of mathematical exploration and uncover the secrets of the domain of f(x) = 2|x-1| + 3. By the end of this guide, you'll have a solid understanding of how to identify the domain of absolute value functions and be well-prepared to tackle more complex mathematical challenges.

Analyzing the Function f(x) = 2|x-1| + 3

When determining the domain of a function, it's essential to first identify any potential restrictions. The function f(x) = 2|x-1| + 3 is an absolute value function, which means it involves the absolute value of an expression. Absolute value functions have a unique property: they return the magnitude of a number, regardless of its sign. In other words, the absolute value of any real number is always non-negative. This characteristic is crucial in understanding why absolute value functions generally have a domain of all real numbers. Let's break down the function f(x) = 2|x-1| + 3 to understand its components and how they contribute to its overall behavior. The expression inside the absolute value, |x-1|, represents the distance between x and 1 on the number line. No matter what value we substitute for x, this distance will always be a non-negative number. Multiplying this distance by 2, as in 2|x-1|, simply scales the distance, but it still remains non-negative. Finally, adding 3 to the scaled distance, as in 2|x-1| + 3, shifts the entire function upwards by 3 units on the y-axis. This vertical shift doesn't affect the domain, as it only changes the output values, not the input values for which the function is defined. Therefore, the key lies in the absolute value expression |x-1|. Since we can take the absolute value of any real number without encountering any mathematical issues, there are no restrictions on the values we can substitute for x. This means that the function f(x) = 2|x-1| + 3 is defined for all real numbers. To further illustrate this point, consider some examples. If we substitute x = 0, we get f(0) = 2|0-1| + 3 = 2(1) + 3 = 5. If we substitute x = 1, we get f(1) = 2|1-1| + 3 = 2(0) + 3 = 3. If we substitute x = -1, we get f(-1) = 2|-1-1| + 3 = 2(2) + 3 = 7. These examples demonstrate that for any value of x we choose, we obtain a valid output. This consistent behavior reinforces the conclusion that the domain of f(x) = 2|x-1| + 3 includes all real numbers. In the next section, we'll formally define the domain and discuss why the other options are incorrect.

Determining the Domain: Why All Real Numbers?

To definitively determine the domain of f(x) = 2|x-1| + 3, we need to consider what values of x would make the function undefined. As we discussed earlier, the only potential restrictions on the domain of a function arise from operations that are not defined for certain values. These operations typically include division by zero, taking the square root of a negative number, or taking the logarithm of a non-positive number. However, in the case of f(x) = 2|x-1| + 3, none of these operations are present. There is no division, no square root, and no logarithm. The only operation involved is the absolute value, which is defined for all real numbers. This means that we can substitute any real number for x in the function f(x) = 2|x-1| + 3 and obtain a valid output. There are no values of x that would cause the function to be undefined. Therefore, the domain of f(x) = 2|x-1| + 3 is the set of all real numbers. This can be expressed mathematically as {x | x ∈ ℝ}, where ℝ represents the set of all real numbers. Now, let's examine why the other options provided are incorrect. Option A, {x | x ≥ 3}, suggests that the domain is limited to values of x greater than or equal to 3. However, as we've established, there are no restrictions on the values of x that can be substituted into the function. We can substitute values less than 3, such as 0, 1, or -1, and obtain valid outputs. Therefore, option A is incorrect. Option B, {x | x < 1}, suggests that the domain is limited to values of x less than 1. Again, this is incorrect because we can substitute values greater than or equal to 1 into the function without encountering any issues. For example, substituting x = 2 gives us f(2) = 2|2-1| + 3 = 2(1) + 3 = 5, which is a valid output. Option C, {x | x ≤ -1}, suggests that the domain is limited to values of x less than or equal to -1. This is also incorrect for the same reasons as options A and B. We can substitute values greater than -1 into the function and obtain valid outputs. For example, substituting x = 0 gives us f(0) = 2|0-1| + 3 = 2(1) + 3 = 5, which is a valid output. In conclusion, the only correct option is D, {x | x = all real numbers}, because the function f(x) = 2|x-1| + 3 is defined for all real numbers. This comprehensive analysis of the function and its components solidifies our understanding of why the domain encompasses all real numbers. In the next section, we'll summarize our findings and reinforce the key concepts discussed.

Conclusion: The Domain of f(x) = 2|x-1| + 3

In summary, the domain of the function f(x) = 2|x-1| + 3 is the set of all real numbers. This is because the function involves the absolute value operation, which is defined for all real numbers, and there are no other operations that would restrict the domain. We have thoroughly analyzed the function, breaking it down into its components and examining how each component contributes to its overall behavior. We have also considered potential restrictions on the domain, such as division by zero, square roots of negative numbers, and logarithms of non-positive numbers, and found that none of these restrictions apply to f(x) = 2|x-1| + 3. Furthermore, we have discussed why the other options provided are incorrect, demonstrating a clear understanding of the domain concept and its application to this specific function. Understanding the domain of a function is a fundamental skill in mathematics, and this article has provided a comprehensive guide to determining the domain of absolute value functions. By grasping the principles discussed here, you will be well-equipped to tackle similar problems and deepen your understanding of mathematical functions. The key takeaway is that when dealing with absolute value functions, you should first consider whether there are any other operations that might restrict the domain. If there are none, then the domain is likely to be all real numbers. This article has not only provided the correct answer but has also explained the reasoning behind it, fostering a deeper understanding of the underlying concepts. This approach is crucial for building a solid foundation in mathematics and developing problem-solving skills that can be applied to a wide range of challenges. As you continue your mathematical journey, remember that understanding the domain of a function is just one piece of the puzzle. There are many other important concepts to explore, such as range, intercepts, asymptotes, and transformations. By mastering these concepts, you will gain a comprehensive understanding of functions and their behavior, enabling you to excel in your mathematical studies. So, keep practicing, keep exploring, and keep pushing your boundaries. The world of mathematics is vast and fascinating, and there is always something new to learn. With dedication and perseverance, you can achieve your mathematical goals and unlock the power of this beautiful and essential subject.

Final Answer: The final answer is (D)