Determine Inequality For Domain Of F(x) = √((1/2)x - 10) + 3

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When venturing into the realm of functions, understanding their domain is paramount. The domain, in essence, delineates the set of all possible input values (x-values) for which the function yields a real output. In this article, we embark on a journey to unravel the domain of the function f(x) = √((1/2)x - 10) + 3. This function, characterized by the presence of a square root, necessitates careful consideration to ensure we only feed it values that result in real numbers. This article will serve as a comprehensive guide to understanding the nuances of domain determination, particularly when dealing with square root functions.

The Core Principle: Non-Negative Radicands

The cornerstone of our analysis lies in the fundamental property of square roots: the radicand (the expression under the square root) must be non-negative to produce a real number. In simpler terms, we cannot take the square root of a negative number and obtain a real result. This principle forms the bedrock of our strategy for identifying the inequality that defines the domain of our function.

For the function f(x) = √((1/2)x - 10) + 3, the radicand is the expression (1/2)x - 10. To ensure that the function produces real outputs, we must impose the condition that this radicand is greater than or equal to zero. This translates to the inequality:

(1/2)x - 10 ≥ 0

This inequality is the key to unlocking the domain of our function. It encapsulates the constraint that the expression under the square root must be non-negative.

Deciphering the Options: A Step-by-Step Analysis

Now, let's dissect the options presented in the question and determine which one aligns with our established principle:

  1. √((1/2)x) ≥ 0: While it is true that the square root of any non-negative number is non-negative, this inequality does not directly address the radicand of our function. It focuses solely on (1/2)x, neglecting the crucial subtraction of 10. Therefore, this option is not the correct representation of the domain constraint.
  2. (1/2)x ≥ 0: Similar to the first option, this inequality omits the essential subtraction of 10 within the square root. It only considers the term (1/2)x, failing to capture the complete radicand. Consequently, this option is not the accurate inequality for determining the domain.
  3. (1/2)x - 10 ≥ 0: This is the precise inequality we derived based on the principle of non-negative radicands. It directly addresses the expression under the square root in our function, ensuring that it remains non-negative. Thus, this option is the correct choice for defining the domain.
  4. √((1/2)x - 10) + 3 ≥ 0: While this inequality is true for all x-values within the domain (since the square root term is always non-negative and we are adding 3), it doesn't directly help us find the domain. It's a consequence of the domain restriction, not the restriction itself. It describes the range of the function (the possible output values), not the domain (the possible input values). It's a true statement but not the one we use to determine the domain.

Solving the Inequality: Unveiling the Domain

Having identified the correct inequality, (1/2)x - 10 ≥ 0, let's proceed to solve it and explicitly determine the domain of the function. We can solve this inequality using basic algebraic manipulations:

  1. Add 10 to both sides: (1/2)x ≥ 10
  2. Multiply both sides by 2: x ≥ 20

This solution reveals 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, we express this domain as [20, ∞).

Visualizing the Domain: A Graphical Perspective

To further solidify our understanding, let's visualize the domain graphically. Imagine a number line. The domain [20, ∞) is represented by a closed bracket at 20 (indicating inclusion) and an arrow extending infinitely to the right. This visual representation provides a clear picture of the allowable input values for our function.

Why Other Options Fail: A Closer Look

It's instructive to revisit why the other options are incorrect. Options 1 and 2, √((1/2)x) ≥ 0 and (1/2)x ≥ 0, respectively, only consider a portion of the radicand. They neglect the crucial subtraction of 10, which significantly impacts the domain. These inequalities would lead to an incorrect domain that includes values for which the radicand becomes negative, resulting in non-real outputs.

Option 4, √((1/2)x - 10) + 3 ≥ 0, although a true statement within the domain, doesn't help us find the domain. It describes the range of the function, not the domain itself. It's a consequence of the domain restriction, not the restriction itself.

Domain in Action: Testing Values

To practically demonstrate the significance of the domain, let's test some x-values. Consider x = 25, which falls within our domain [20, ∞). Plugging this value into our function:

f(25) = √((1/2)(25) - 10) + 3 = √(12.5 - 10) + 3 = √2.5 + 3, which yields a real number.

Now, let's try x = 15, a value outside our domain:

f(15) = √((1/2)(15) - 10) + 3 = √(7.5 - 10) + 3 = √(-2.5) + 3, which results in the square root of a negative number, indicating a non-real output.

These examples vividly illustrate how the domain dictates the permissible input values for the function to produce real outputs.

Domains in the Real World: Applications

The concept of domain extends far beyond mathematical exercises; it plays a crucial role in various real-world applications. Consider scenarios involving physical quantities like distance, time, or volume. These quantities cannot be negative, thereby imposing constraints on the domain of functions that model these scenarios. For instance, if a function represents the distance traveled by a car over time, the domain would typically be restricted to non-negative time values.

Mastering Domain: A Summary of Key Concepts

Let's consolidate the key concepts we've explored:

  • The domain of a function is the set of all possible input values (x-values) that produce real outputs.
  • For square root functions, the radicand (the expression under the square root) must be non-negative.
  • The inequality that defines the domain is obtained by setting the radicand greater than or equal to zero.
  • Solving this inequality yields the set of permissible x-values.
  • The domain can be visualized graphically on a number line.
  • The domain has practical implications in real-world applications where physical quantities impose constraints.

Conclusion: The Importance of Domain Determination

In conclusion, determining the domain of a function is a fundamental aspect of mathematical analysis. For functions involving square roots, the principle of non-negative radicands is paramount. By carefully analyzing the radicand and constructing the appropriate inequality, we can effectively unveil the domain and gain a comprehensive understanding of the function's behavior. This understanding is not only crucial for mathematical rigor but also for applying functions in real-world contexts where constraints and limitations exist.

Therefore, the correct inequality to find the domain of f(x) = √((1/2)x - 10) + 3 is (1/2)x - 10 ≥ 0. This inequality ensures that the expression under the square root remains non-negative, guaranteeing real outputs for the function.

Understanding domain restrictions is paramount in mathematics, especially when dealing with functions like the one presented. It's not just about solving for x; it's about understanding the limitations and boundaries within which our mathematical models operate. This concept permeates various fields, from physics and engineering to economics and computer science, making it a foundational skill for anyone engaging with quantitative analysis.

The function f(x) = √((1/2)x - 10) + 3 serves as an excellent example for illustrating this principle. The square root function, a cornerstone of algebra and calculus, inherently demands that its argument (the radicand) be non-negative. This seemingly simple constraint opens the door to a deeper understanding of mathematical rigor and the importance of considering the validity of operations within a given context.

Moreover, the constant addition of 3 outside the square root does not impact the domain; it only affects the range (the set of output values). This distinction is crucial for differentiating between input restrictions (domain) and output behavior (range). It is essential to recognize which components of a function influence which aspect of its behavior.

The process of solving the inequality (1/2)x - 10 ≥ 0 is a prime example of applying algebraic manipulation to uncover crucial information about a function. By isolating x, we transform the inequality into a readily interpretable form: x ≥ 20. This straightforward solution unveils the entire domain, elegantly capturing the permissible input values.

In essence, the question posed isn't merely about finding the right inequality; it's about fostering a deeper appreciation for the nuances of mathematical functions and the critical role of domain restrictions. It’s about understanding that mathematical expressions are not just abstract symbols but rather representations of relationships that operate within defined boundaries. And mastering this understanding is key to unlocking the power of mathematics in solving real-world problems.