Finding The Domain Of F(x) = √(1/2 X - 10 + 3) A Step-by-Step Guide

#h1 Understanding the Domain of Functions: A Deep Dive into f(x) = √(1/2 x - 10 + 3)

In the realm of mathematics, functions reign supreme, acting as fundamental building blocks for modeling real-world phenomena. To truly grasp the essence of a function, we must delve into its domain, which represents the set of all permissible input values that the function can accept without venturing into mathematical impossibilities. This article embarks on an in-depth exploration of determining the domain of a specific function, $f(x) = \sqrt{\frac{1}{2} x - 10 + 3}$, while dissecting the underlying principles that govern domain restrictions. Our focus centers on the crucial role of the square root function and its inherent demand for non-negative inputs.

The Essence of Domain: Permissible Input Values

The domain of a function serves as a gatekeeper, dictating the set of input values for which the function yields a valid output. Think of it as the function's permissible operating range. Certain mathematical operations impose restrictions on the domain, and the square root is a prime example. The square root function, denoted by $\sqrt{x}$, demands that its input, $x$, be greater than or equal to zero. This stems from the fact that the square root of a negative number ventures into the realm of imaginary numbers, which fall outside the scope of real-valued functions. To understand the nuances of domain restrictions, let's begin by examining the key concept of permissible input values within the context of mathematical functions.

When dealing with functions, the domain is a crucial aspect that defines the set of all possible input values (often represented as 'x') for which the function will produce a valid output. In simpler terms, it's the range of values you can plug into a function without causing mathematical errors or undefined results. Understanding the domain is essential because it helps us interpret the function's behavior and ensures that our calculations are meaningful. For instance, consider the function $f(x) = \frac{1}{x}$. Here, the domain excludes $x = 0$ because division by zero is undefined. Similarly, the square root function, $f(x) = \sqrt{x}$, requires non-negative inputs since the square root of a negative number is not a real number.

Now, let's delve deeper into the specifics of finding the domain for the function $f(x) = \sqrt{\frac{1}{2} x - 10 + 3}$. The presence of a square root immediately alerts us to a potential domain restriction. The expression inside the square root, also known as the radicand, must be greater than or equal to zero. This is because the square root of a negative number is not defined in the set of real numbers. Therefore, to determine the domain of this function, we need to identify the values of $x$ that make the radicand non-negative. This involves setting up an inequality and solving for $x$, which will give us the range of permissible input values. The importance of identifying these restrictions cannot be overstated, as they directly influence the function's behavior and its applicability in real-world scenarios. By carefully considering these limitations, we can ensure that our mathematical models accurately reflect the situations they are intended to represent.

The Square Root's Demand: Non-Negative Inputs

The square root function, a cornerstone of mathematical operations, possesses a unique characteristic that dictates a specific condition for its domain: the input must be non-negative. This restriction arises from the fundamental definition of the square root, which seeks a number that, when multiplied by itself, yields the original input. Consider the square root of 9, denoted as $\sqrt{9}$. The answer is 3 because 3 multiplied by itself (3 * 3) equals 9. However, when we delve into negative numbers, this concept encounters a hurdle. There is no real number that, when multiplied by itself, results in a negative number. For instance, if we attempt to find the square root of -9, we encounter the realm of imaginary numbers, which are beyond the scope of typical real-valued functions. To emphasize this point further, think about squaring any real number. Whether the number is positive or negative, squaring it always results in a non-negative value. For example, $3^2 = 9$ and $(-3)^2 = 9$. This inherent property of squaring dictates that the reverse operation, the square root, can only accept non-negative inputs to produce real number outputs.

The practical implication of this restriction is that whenever a function involves a square root, we must ensure that the expression inside the square root (the radicand) is greater than or equal to zero. This requirement forms the basis for determining the domain of many functions, particularly those that model physical phenomena or real-world scenarios. In various applications, such as physics and engineering, functions often represent measurable quantities that cannot be negative, such as distance, time, or mass. Therefore, the square root's demand for non-negative inputs aligns perfectly with the constraints of these real-world contexts. Ignoring this restriction can lead to nonsensical results or mathematical inconsistencies. For example, if we were to model the speed of an object using a function involving a square root, a negative input would imply an imaginary speed, which is physically impossible. Thus, understanding and adhering to the square root's domain restriction is crucial for both mathematical accuracy and the meaningful interpretation of results.

Cracking the Code: Finding the Domain of f(x) = √(1/2 x - 10 + 3)

Now, let's put our knowledge into practice and determine the domain of the function $f(x) = \sqrt{\frac{1}{2} x - 10 + 3}$. As we've established, the key lies in ensuring that the expression inside the square root, the radicand, is greater than or equal to zero. In this case, the radicand is $\frac{1}{2} x - 10 + 3$. Our mission is to find the values of $x$ that satisfy the inequality: $\frac{1}{2} x - 10 + 3 \geq 0$. This inequality encapsulates the very essence of the domain restriction imposed by the square root function. By solving this inequality, we'll unveil the permissible values of $x$ that allow the function to produce real-valued outputs.

The process of solving this inequality involves a series of algebraic manipulations aimed at isolating $x$. The first step is to simplify the expression by combining the constant terms: $-10 + 3 = -7$. This gives us the simplified inequality: $\frac{1}{2} x - 7 \geq 0$. Next, we want to isolate the term involving $x$. We can achieve this by adding 7 to both sides of the inequality: $\frac{1}{2} x - 7 + 7 \geq 0 + 7$, which simplifies to $\frac{1}{2} x \geq 7$. Finally, to solve for $x$, we multiply both sides of the inequality by 2: $2 * \frac{1}{2} x \geq 2 * 7$, resulting in $x \geq 14$. This inequality, $x \geq 14$, represents the domain of the function $f(x) = \sqrt{\frac{1}{2} x - 10 + 3}$. It tells us that the function is defined for all values of $x$ that are greater than or equal to 14. Any value of $x$ less than 14 would result in a negative radicand, leading to an undefined result in the real number system. Therefore, the solution to the inequality, $x \geq 14$, is the key to unlocking the domain of our function.

Deciphering the Options: Which Inequality Reigns Supreme?

Having conquered the challenge of finding the domain, let's now turn our attention to the multiple-choice options presented. Our goal is to identify the inequality that accurately reflects the domain of $f(x) = \sqrt{\frac{1}{2} x - 10 + 3}$. We've already determined that the domain is defined by the inequality $x \geq 14$. The task at hand is to match this solution with one of the given options.

• Option A: $\sqrt{\frac{1}{2} x} \geq 0$ This inequality focuses solely on the $\frac{1}{2}x$ term under a square root, neglecting the crucial constants -10 and +3. While it correctly acknowledges the non-negativity requirement of a square root, it doesn't capture the entire radicand of our function. Therefore, this option is not the precise representation of our function's domain restriction.

• Option B: $\frac{1}{2} x \geq 0$ This option, like option A, isolates a portion of the radicand, specifically $\frac{1}{2}x$, and sets it greater than or equal to zero. It omits the essential constants -10 and +3, which significantly impact the domain. Solving this inequality would yield $x \geq 0$, a domain much broader than the actual domain of our function. Hence, option B is not the correct choice.

• Option C: $\frac{1}{2} x - 10 \geq 0$ This option comes tantalizingly close to the correct answer. It includes the $\frac{1}{2}x$ term and the -10 constant, but it omits the +3. While it's a step closer than options A and B, the missing +3 prevents it from accurately representing the domain restriction. If we were to solve this inequality, we would obtain $x \geq 20$, which is not the true domain of our function.

To solve the initial problem, we need to find the inequality that accurately represents the condition for the domain of the function $f(x) = \sqrt{\frac{1}{2} x - 10 + 3}$. The domain of a square root function is the set of all $x$ values for which the expression inside the square root is greater than or equal to zero. Therefore, the correct inequality must reflect this condition.

In the given function, the expression inside the square root is $\frac{1}{2} x - 10 + 3$. To find the domain, we need to set this expression greater than or equal to zero:

$$\frac{1}{2} x - 10 + 3 \geq 0$$

This inequality ensures that the value inside the square root is non-negative, which is a requirement for the function to be defined in the real number system. Now, let's examine the provided options to see which one correctly represents this condition.

• Option A: $\sqrt{\frac{1}{2} x} \geq 0$ This inequality is true for $x \geq 0$, but it does not account for the entire expression inside the square root in the original function. It only considers a part of it, thus not accurately representing the domain restriction.

• Option B: $\frac{1}{2} x \geq 0$ Similar to option A, this inequality is valid for $x \geq 0$, but it fails to include the constants -10 and +3 from the original expression inside the square root. Therefore, it does not fully capture the domain requirement of the function.

• Option C: $\frac{1}{2} x - 10 \geq 0$ This option includes the $\frac{1}{2} x$ term and the -10 constant, but it misses the +3. It's closer to the correct inequality than options A and B, but it's still not the precise representation of the domain restriction.

To find the correct inequality, we need to simplify the expression inside the square root in the original function:

$$\frac{1}{2} x - 10 + 3 = \frac{1}{2} x - 7$$

So, the inequality that correctly represents the domain restriction should be:

$$\frac{1}{2} x - 7 \geq 0$$

This inequality ensures that the entire expression inside the square root is non-negative. Now, let's compare this with the given options again.

Upon closer inspection, we realize that none of the provided options perfectly matches the inequality we derived. However, if we were to choose the option that is closest to the correct representation, it would be Option C: $\frac{1}{2} x - 10 \geq 0$. This option is the most similar because it includes the $\frac{1}{2} x$ term and one of the constants, -10. The only missing part is the +3. Despite this, it's the best choice among the given options.

In summary, the inequality that can be used to find the domain of $f(x) = \sqrt{\frac{1}{2} x - 10 + 3}$ is best represented by Option C, although it's not a perfect match. The ideal inequality should be $\frac{1}{2} x - 7 \geq 0$.

Conclusion: The Domain Decoded

In this comprehensive exploration, we've dissected the process of determining the domain of a square root function, specifically $f(x) = \sqrt{\frac{1}{2} x - 10 + 3}$. We've emphasized the critical role of the square root's non-negativity requirement and demonstrated how to translate this requirement into an inequality. By solving this inequality, we unveiled the function's domain, the set of all permissible input values. While the provided options didn't perfectly align with our solution, we identified the closest representation, highlighting the importance of a thorough understanding of domain restrictions. Mastering the art of domain determination is paramount for a solid foundation in mathematics, enabling us to navigate the intricacies of functions and their real-world applications with confidence.