Finding The Domain Of A Square Root Function F(x) = √((1/2)x - 10) + 3

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In mathematics, understanding the domain of a function is crucial for analyzing its behavior and properties. The domain represents the set of all possible input values (x-values) for which the function produces a valid output (y-value). When dealing with functions involving square roots, we encounter a specific constraint: the expression under the square root must be non-negative. This is because the square root of a negative number is not a real number. In this article, we will explore how to determine the domain of a function involving a square root, using the example of f(x) = √((1/2)x - 10) + 3. By the end of this guide, you'll be equipped with the knowledge and skills to confidently identify the appropriate inequality for finding the domain of similar functions.

The Core Concept: Non-Negative Radicand

The key to finding the domain of a function with a square root lies in understanding the radicand, which is the expression under the square root symbol. For the function to produce real number outputs, the radicand must be greater than or equal to zero. This principle forms the foundation for determining the domain. When we have square root functions, we must ensure that the expression inside the square root, known as the radicand, is non-negative. This is because the square root of a negative number is not defined within the realm of real numbers. To illustrate, consider the function f(x) = √(x). The domain of this function is all non-negative real numbers, represented as x ≥ 0. If we were to input a negative value, such as x = -1, we would encounter √(-1), which is an imaginary number. This underscores the importance of ensuring a non-negative radicand when dealing with square root functions.

Applying the Concept to Our Function

Now, let's apply this concept to our specific function, f(x) = √((1/2)x - 10) + 3. The radicand in this case is (1/2)x - 10. To ensure that the function produces real number outputs, we need to set up an inequality that guarantees the radicand is non-negative. This means we need to find the values of x for which (1/2)x - 10 ≥ 0. The "+ 3" outside the square root does not affect the domain because it simply shifts the entire function vertically and does not introduce any restrictions on the possible x-values. Therefore, the crucial part of the function that determines the domain is the square root term, specifically the radicand. By focusing on the radicand, we can establish the necessary inequality to solve for the domain of the function. This involves isolating x in the inequality, which will give us the range of x-values that make the radicand non-negative.

Why Other Options Are Incorrect

Let's examine why the other inequality options provided are not suitable for finding the domain of f(x):

  1. √(1/2)x ≥ 0: This inequality only considers the square root of a portion of the radicand. It neglects the crucial "- 10" term, which significantly impacts the domain. The term inside the square root is (1/2)x - 10, not just (1/2)x. Ignoring the "- 10" would lead to an incorrect domain because it doesn't account for the shift caused by this constant term. For example, if we solved √(1/2)x ≥ 0, we would get x ≥ 0, but this doesn't ensure that the entire radicand (1/2)x - 10 is non-negative.

  2. (1/2)x ≥ 0: Similar to the previous option, this inequality omits the "- 10" term and fails to address the requirement that the entire radicand must be non-negative. Although it's a step closer because it considers the variable x within the radicand, it still misses the critical component that shifts the domain. Solving (1/2)x ≥ 0 gives x ≥ 0, which, as mentioned before, isn't the complete solution for the domain of the function.

  3. √(1/2)x - 10 + 3 ≥ 0: This inequality includes the entire function, but it's not the correct approach for finding the domain. The inequality implies that the entire function's output must be non-negative, which is a condition related to the range of the function, not its domain. The domain is concerned with the input values (x-values) that make the function valid, not the output values (y-values). This option mixes the concept of domain and range, leading to an incorrect understanding of how to find the domain of the function.

The Correct Inequality: (1/2)x - 10 ≥ 0

As we've established, the correct inequality to find the domain of f(x) = √((1/2)x - 10) + 3 is (1/2)x - 10 ≥ 0. This inequality directly addresses the requirement that the radicand, the expression under the square root, must be non-negative for the function to produce real number outputs. It accurately captures the constraint imposed by the square root function and provides the foundation for solving for the valid input values (x-values).

Solving the Inequality

To find the domain, we need to solve the inequality (1/2)x - 10 ≥ 0. Let's walk through the steps:

  1. Add 10 to both sides: This isolates the term with x on one side of the inequality. Adding 10 to both sides gives us: (1/2)x ≥ 10.

  2. Multiply both sides by 2: This eliminates the fraction and solves for x. Multiplying both sides by 2 yields: x ≥ 20.

The solution to the inequality is x ≥ 20. This means that the domain of the function f(x) = √((1/2)x - 10) + 3 consists of all real numbers greater than or equal to 20. In interval notation, the domain is [20, ∞). This interval includes 20 because the inequality is inclusive (≥), and it extends to infinity, representing all real numbers greater than 20. Understanding how to solve the inequality provides a clear picture of the function's permissible input values.

Visualizing the Domain

To further solidify our understanding, let's visualize the domain. We can represent the domain graphically on a number line. Draw a number line and mark the point 20. Since the domain includes all numbers greater than or equal to 20, we draw a closed circle at 20 (indicating that 20 is included) and shade the line to the right, extending towards positive infinity. This visual representation clearly illustrates that any x-value less than 20 would result in a negative radicand, making the function undefined in the real number system. The visualization reinforces the concept that the domain is the set of all x-values for which the function produces a real number output.

Testing Values

We can test values within and outside the domain to confirm our solution. Let's try x = 25, which is within the domain:

f(25) = √((1/2)(25) - 10) + 3 = √(12.5 - 10) + 3 = √(2.5) + 3

This gives us a real number output, confirming that 25 is indeed in the domain.

Now, let's try x = 15, which is outside the domain:

f(15) = √((1/2)(15) - 10) + 3 = √(7.5 - 10) + 3 = √(-2.5) + 3

This results in the square root of a negative number, indicating that 15 is not in the domain. The testing of values helps validate the solution and reinforces the understanding of the domain's boundaries.

Conclusion: Mastering Domain Determination

In this comprehensive guide, we've explored the process of determining the domain of a function involving a square root, using the example of f(x) = √((1/2)x - 10) + 3. We've established that the key is to ensure the radicand is non-negative, leading us to the correct inequality: (1/2)x - 10 ≥ 0. By solving this inequality, we found that the domain of the function is x ≥ 20, represented in interval notation as [20, ∞). Understanding how to find the domain of functions is a fundamental skill in mathematics. It allows us to analyze the behavior of functions, identify their limitations, and solve related problems accurately. By mastering the concept of non-negative radicands and applying the steps outlined in this guide, you can confidently determine the domain of various functions involving square roots. Remember, the domain is the foundation upon which the rest of the function's behavior is built, making its accurate determination essential for further analysis and application.

This knowledge not only helps in solving mathematical problems but also in understanding the real-world applications of functions, where domains represent the practical limitations of a given situation. Keep practicing with different functions, and you'll become proficient in identifying their domains with ease.