Domain Of F(x) = √(3x - 27) Explained With Examples

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In the realm of mathematics, understanding the domain of a function is crucial for grasping its behavior and applicability. The domain of a function encompasses all possible input values (often represented by 'x') for which the function produces a valid output. In simpler terms, it's the set of all 'x' values that you can plug into the function without encountering any mathematical errors, such as division by zero or taking the square root of a negative number. When dealing with different types of functions, the methods for finding the domain may vary, and this exploration focuses on square root functions, which present a unique challenge. Square root functions, characterized by the presence of a radical symbol (√), introduce a specific constraint: the expression under the square root must be greater than or equal to zero. This restriction arises because the square root of a negative number is not defined within the realm of real numbers. Therefore, when determining the domain of a square root function, this fundamental principle becomes the guiding rule. Let's embark on a journey to unravel the intricacies of finding the domain, with a specific focus on the function f(x) = √(3x - 27), a quintessential example that beautifully illustrates the concepts involved.

Deciphering the Square Root Constraint

At the heart of finding the domain of the function f(x) = √(3x - 27) lies a fundamental principle: the expression under the square root, known as the radicand, must be greater than or equal to zero. This principle stems from the very definition of the square root function within the realm of real numbers. We know that the square root of a negative number is not a real number, as there is no real number that, when multiplied by itself, yields a negative result. For example, √-4 is not a real number because there is no real number that equals -4 when squared. This inherent constraint dictates our approach to determining the domain of square root functions. In the context of f(x) = √(3x - 27), the radicand is the expression 3x - 27. To ensure that the output of the function is a real number, we must guarantee that 3x - 27 is non-negative. This transforms our problem into an inequality: 3x - 27 ≥ 0. Solving this inequality will unveil the set of all x values that satisfy the condition, thereby revealing the domain of our function. By adhering to this principle, we navigate the complexities of square root functions and ensure that we are working within the boundaries of real number mathematics. Understanding this constraint is not just a mathematical technicality; it's a gateway to comprehending the function's behavior and its real-world applications. It dictates the input values for which the function yields meaningful outputs, shaping our understanding of the function's scope and limitations.

Solving the Inequality: A Step-by-Step Approach

Now that we've established the crucial inequality 3x - 27 ≥ 0, we embark on the process of solving it to determine the valid range of x values for our function f(x) = √(3x - 27). Solving inequalities follows a similar methodology to solving equations, but with a key difference: we must be mindful of how operations affect the inequality sign. Our goal is to isolate x on one side of the inequality, gradually simplifying the expression until we arrive at a solution in the form x ≥ a or x ≤ a, where a is a constant. The first step in solving 3x - 27 ≥ 0 is to eliminate the constant term on the left side. We achieve this by adding 27 to both sides of the inequality. This maintains the balance of the inequality while moving us closer to isolating x. Adding 27 to both sides yields 3x ≥ 27. Next, we need to isolate x completely. Since x is currently multiplied by 3, we perform the inverse operation: division. We divide both sides of the inequality by 3. It's important to note that when dividing or multiplying an inequality by a negative number, we must reverse the inequality sign. However, in this case, we are dividing by a positive number (3), so the inequality sign remains unchanged. Dividing both sides by 3 gives us x ≥ 9. This is our solution! It tells us that the function f(x) = √(3x - 27) is defined for all values of x that are greater than or equal to 9. This solution encapsulates the essence of the domain of the function, providing a clear and concise answer to our quest.

Expressing the Domain: Interval Notation and Graphical Representation

With the solution to the inequality in hand (x ≥ 9), we now have a precise understanding of the domain of the function f(x) = √(3x - 27). However, to fully convey this understanding, we need to express the domain in a standard mathematical notation and visualize it graphically. One of the most common and convenient ways to represent the domain is through interval notation. Interval notation uses brackets and parentheses to indicate the range of values that x can take. A square bracket ([ or ]) indicates that the endpoint is included in the interval, while a parenthesis (( or )) indicates that the endpoint is excluded. Since our solution is x ≥ 9, this means that x can be 9 or any number greater than 9. In interval notation, this is represented as [9, ∞). The square bracket on the 9 indicates that 9 is included in the domain, and the parenthesis on the infinity symbol (∞) indicates that infinity is not a specific number but rather an unbounded concept. This notation elegantly captures the idea that the domain extends indefinitely in the positive direction. Complementing interval notation, a graphical representation of the domain provides a visual understanding of the valid x values. We can represent the domain on a number line. Draw a number line and mark the point 9. Since 9 is included in the domain, we use a closed circle (or a filled-in dot) at 9. Then, we draw an arrow extending from 9 to the right, indicating that all values greater than 9 are also included in the domain. This visual representation reinforces the concept that the domain consists of all x values from 9 onwards, encompassing the infinite expanse of numbers to the right. By combining interval notation and graphical representation, we achieve a comprehensive and intuitive understanding of the domain of the function f(x) = √(3x - 27).

Connecting the Solution to the Options

Having meticulously determined the domain of the function f(x) = √(3x - 27), it's time to connect our findings to the provided options. Our analysis led us to the solution x ≥ 9, which we expressed in interval notation as [9, ∞). Now, let's examine the given options:

  • A. (-9, ∞)
  • B. (9, ∞)
  • C. [-9, ∞)
  • D. [9, ∞)

By carefully comparing our derived domain [9, ∞) with the options, we can confidently identify the correct answer. Option A, (-9, ∞), suggests that the domain includes all values greater than -9, which is incorrect as it includes values less than 9. Option B, (9, ∞), is close but omits the crucial inclusion of 9 itself in the domain. The parenthesis indicates that 9 is not included, which contradicts our solution x ≥ 9. Option C, [-9, ∞), is also incorrect as it includes values less than 9, specifically starting from -9. Option D, [9, ∞), perfectly matches our derived domain. The square bracket correctly indicates that 9 is included, and the infinity symbol signifies the unbounded extension of the domain to positive infinity. Therefore, the correct answer is unequivocally Option D. This process of connecting the solution to the options underscores the importance of a thorough and accurate determination of the domain. By meticulously following the steps, from establishing the inequality to solving it and expressing the solution in interval notation, we ensure that we can confidently select the correct answer from a set of choices. The ability to connect theoretical calculations to practical answer selection is a cornerstone of mathematical problem-solving.

Conclusion: The Domain Unveiled

In this comprehensive exploration, we embarked on a journey to unravel the domain of the function f(x) = √(3x - 27). We began by establishing the fundamental principle that the radicand (the expression under the square root) must be greater than or equal to zero. This principle guided our subsequent steps, transforming the problem into solving the inequality 3x - 27 ≥ 0. Through a step-by-step approach, we successfully isolated x, arriving at the solution x ≥ 9. This solution, the heart of the domain, signifies that the function is defined for all values of x greater than or equal to 9. To express this domain in a standard mathematical language, we employed interval notation, representing it as [9, ∞). This notation concisely captures the essence of the domain, indicating that 9 is included and the range extends indefinitely towards positive infinity. Furthermore, we visualized the domain graphically on a number line, solidifying our understanding of the valid x values. Finally, we connected our derived domain to the provided options, confidently identifying Option D, [9, ∞), as the correct answer. This exercise highlights the importance of understanding the constraints imposed by mathematical functions, particularly square root functions. The domain is not merely a technicality; it's a fundamental aspect of a function that dictates its behavior and applicability. By mastering the techniques for finding the domain, we gain a deeper appreciation for the intricacies of mathematics and its power to model real-world phenomena. The quest to unveil the domain of f(x) = √(3x - 27) has been a journey of discovery, culminating in a clear and comprehensive understanding of the function's permissible input values.