Finding The Domain Of A Square Root Function F(x) = √(x-14)

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Delving into the realm of mathematical functions, one encounters various types, each with its unique characteristics and properties. Among these, square root functions stand out due to their inherent restriction on the input values, known as the domain. The domain of a function encompasses all possible input values for which the function produces a valid output. In the case of square root functions, this restriction stems from the fact that the square root of a negative number is not defined within the set of real numbers. Understanding the domain of a square root function is crucial for solving equations, graphing functions, and applying mathematical concepts in real-world scenarios.

To understand which value of x is in the domain of f(x) = √(x - 14), we need to have a solid grasp of what a function's domain actually means, especially in the context of square root functions. The domain of a function is essentially the set of all possible input values (often x values) for which the function will produce a valid output. For square root functions like the one given, the key constraint is that we cannot take the square root of a negative number and still obtain a real number as a result. This restriction arises from the very definition of the square root operation within the real number system. Recall that the square root of a number y is a value that, when multiplied by itself, equals y. For positive numbers, this is straightforward. For instance, the square root of 9 is 3 because 3 multiplied by 3 equals 9. However, when we consider negative numbers, there is no real number that, when multiplied by itself, yields a negative result. This is because the product of two positive numbers is positive, and the product of two negative numbers is also positive. It is this fundamental mathematical principle that dictates the domain restriction for square root functions.

Therefore, for the function f(x) = √(x - 14), the expression inside the square root, which is (x - 14), must be greater than or equal to zero. This is because taking the square root of a negative number would result in an imaginary number, which falls outside the scope of real-valued functions. To determine the valid values of x, we set up the following inequality: x - 14 ≥ 0. Solving this inequality will provide us with the range of x values that constitute the domain of the function. To solve the inequality, we simply add 14 to both sides, resulting in x ≥ 14. This inequality tells us that the domain of the function f(x) = √(x - 14) consists of all real numbers x that are greater than or equal to 14. In interval notation, we can express the domain as [14, ∞), where the square bracket indicates that 14 is included in the domain, and the parenthesis indicates that infinity is not a specific number but rather an unbounded concept. Now, let's analyze the given options to see which value of x falls within this domain.

Analyzing the Options: Identifying the Valid x Value

Now, we'll examine each of the provided options to determine which one satisfies the condition x ≥ 14, ensuring that the value of x lies within the function's domain.

  • A. x = 0: This value is clearly less than 14. When we substitute x = 0 into the expression inside the square root, we get 0 - 14 = -14. Taking the square root of -14 would result in an imaginary number, so x = 0 is not in the domain of the function.
  • B. x = 20: This value is greater than 14. Substituting x = 20 into the expression inside the square root gives us 20 - 14 = 6. The square root of 6 is a real number, approximately 2.449, so x = 20 is a valid input value and lies within the domain of the function.
  • C. x = 13: This value is less than 14. Substituting x = 13 into the expression inside the square root gives us 13 - 14 = -1. The square root of -1 would result in an imaginary number, so x = 13 is not in the domain of the function.
  • D. x = -1: This value is also less than 14. Substituting x = -1 into the expression inside the square root gives us -1 - 14 = -15. The square root of -15 would result in an imaginary number, so x = -1 is not in the domain of the function.

Based on this analysis, we can conclude that only x = 20 satisfies the condition x ≥ 14 and therefore lies within the domain of the function f(x) = √(x - 14).

Conclusion: The Domain Unveiled

In conclusion, the domain of the function f(x) = √(x - 14) is the set of all real numbers x such that x ≥ 14. Among the given options, only the value x = 20 falls within this domain, making it the correct answer. Understanding the concept of a function's domain, particularly for square root functions, is crucial for mathematical problem-solving and real-world applications. By recognizing the restriction imposed by the square root operation, we can accurately determine the valid input values for a given function and ensure that the output is a real number. This understanding not only helps in solving mathematical problems but also enhances our ability to model and analyze various phenomena in fields such as physics, engineering, and economics, where square root functions frequently appear.

Furthermore, the principles discussed here extend beyond just square root functions. Similar domain restrictions apply to other types of functions, such as logarithmic functions and rational functions, where specific conditions must be met to ensure valid outputs. For instance, the argument of a logarithmic function must be strictly positive, and the denominator of a rational function cannot be zero. Recognizing and understanding these restrictions is a fundamental aspect of mathematical literacy and is essential for anyone working with mathematical models and equations. Therefore, mastering the concept of a function's domain is a valuable skill that transcends specific mathematical problems and contributes to a broader understanding of mathematical principles and their applications.

Therefore, the answer is B. x = 20.