Finding The Domain Of Y=4√(4x+2) A Step-by-Step Guide

In mathematics, determining the domain of a function is a fundamental concept. The domain represents the set of all possible input values (often x-values) for which the function produces a valid output (often y-values). For functions involving square roots, like the one presented, the domain is restricted by the fact that we cannot take the square root of a negative number in the real number system. This article will thoroughly explore how to find the domain of the function $y = 4\sqrt{4x + 2}$, providing a step-by-step explanation to ensure a clear understanding. Understanding the domain is crucial for graphing functions, solving equations, and various other mathematical applications. The domain gives us the boundaries within which the function is defined and behaves predictably. Let's delve into the process of finding the domain, which involves identifying any restrictions on the input variable x that would result in an undefined or non-real output. We will address the specific function at hand, breaking down each step to determine the range of x values that make the function valid. By the end of this article, you'll not only know the domain of this function but also grasp the general method for finding the domain of similar functions involving square roots. This knowledge will serve as a solid foundation for more advanced mathematical concepts and problem-solving scenarios. So, let's embark on this mathematical journey and uncover the domain of the function $y = 4\sqrt{4x + 2}$.

Understanding the Domain of a Function

To find the domain of the function $y = 4\sqrt{4x + 2}$, we must first understand the fundamental concept of a function's domain. The domain of a function is the set of all possible input values (x-values) that will produce a valid output (y-value). In simpler terms, it's the range of x-values for which the function is defined. When dealing with square root functions, there's a critical restriction: the expression inside the square root (the radicand) must be greater than or equal to zero. This is because the square root of a negative number is not a real number. Therefore, the domain of a square root function is determined by setting the radicand greater than or equal to zero and solving for x. For instance, if we have a function like $y = \sqrt{f(x)}$, the domain is found by solving the inequality $f(x) \geq 0$. This inequality ensures that the expression inside the square root is non-negative, leading to a real-valued output. The coefficient outside the square root, in this case, 4, does not affect the domain. It only affects the range of the function, which is the set of all possible output values. Understanding these principles is essential for finding the domain of our given function and similar mathematical problems. This concept is not just limited to square root functions; other types of functions, such as rational functions (where the denominator cannot be zero) and logarithmic functions (where the argument must be positive), also have specific restrictions that define their domains. So, with this foundational knowledge, we are now well-equipped to find the domain of $y = 4\sqrt{4x + 2}$ by focusing on the radicand and ensuring it remains non-negative.

Step-by-Step Solution for $y=4\sqrt{4x+2}$

To determine the domain of the function $y = 4\sqrt{4x + 2}$, we need to focus on the expression inside the square root, which is $4x + 2$. As we established earlier, the radicand (the expression inside the square root) must be greater than or equal to zero to ensure a real number result. Therefore, we set up the inequality: $4x + 2 \geq 0$. Now, we need to solve this inequality for x. First, we subtract 2 from both sides of the inequality: $4x + 2 - 2 \geq 0 - 2$, which simplifies to $4x \geq -2$. Next, we divide both sides by 4 to isolate x: $\frac{4x}{4} \geq \frac{-2}{4}$, which simplifies to $x \geq -\frac{1}{2}$. This result, $x \geq -\frac{1}{2}$, defines the domain of the function. It means that any value of x greater than or equal to $-\frac{1}{2}$ will produce a real number output for the function $y = 4\sqrt{4x + 2}$. Values of x less than $-\frac{1}{2}$ would result in a negative value inside the square root, leading to an imaginary number, which is not in the real number system. In summary, the domain of the function is all real numbers x such that $x$ is greater than or equal to $-\frac{1}{2}$. This step-by-step solution clearly illustrates how to approach finding the domain of functions involving square roots, emphasizing the importance of ensuring the radicand is non-negative. Understanding this process is crucial for further studies in algebra and calculus, where domains play a significant role in analyzing functions.

Detailed Explanation of the Solution

Let's delve deeper into the solution to ensure a comprehensive understanding. We've established that the domain of $y = 4\sqrt{4x + 2}$ is determined by the inequality $4x + 2 \geq 0$. This inequality arises from the fundamental requirement that the value inside a square root cannot be negative within the realm of real numbers. Now, let’s break down each step in solving this inequality. We start with $4x + 2 \geq 0$. The first step is to isolate the term containing x. To do this, we subtract 2 from both sides of the inequality. This maintains the balance of the inequality, similar to how adding or subtracting the same value from both sides of an equation keeps it balanced. Subtracting 2 gives us $4x \geq -2$. The next step is to isolate x completely. Since x is multiplied by 4, we divide both sides of the inequality by 4. This gives us $x \geq -\frac{2}{4}$. Simplifying the fraction $-\frac{2}{4}$, we get $x \geq -\frac{1}{2}$. This final inequality, $x \geq -\frac{1}{2}$, represents the domain of the function. It tells us that the function is defined for all x-values that are greater than or equal to $-\frac{1}{2}$. Any x-value less than $-\frac{1}{2}$ would make the expression inside the square root negative, resulting in a non-real number. For example, if we substitute $x = -1$ into the expression $4x + 2$, we get $4(-1) + 2 = -4 + 2 = -2$, which is negative. This confirms that x-values less than $-\frac{1}{2}$ are not in the domain. On the other hand, if we substitute $x = 0$, we get $4(0) + 2 = 2$, which is positive, and if we substitute $x = -\frac{1}{2}$, we get $4(-\frac{1}{2}) + 2 = -2 + 2 = 0$, which is also valid. This thorough explanation clarifies why the solution $x \geq -\frac{1}{2}$ is the correct domain for the function. It reinforces the importance of understanding the restrictions imposed by mathematical operations like square roots when determining the domain of a function.

Identifying the Correct Option

Now that we have determined the domain of the function $y = 4\sqrt{4x + 2}$ to be $x \geq -\frac{1}{2}$, we can confidently identify the correct option among the given choices. The options are:

A. $x \geq \frac{1}{2}$ B. $x \geq -2$ C. $x \geq -\frac{1}{2}$ D. $x > \frac{1}{2}$

Comparing our solution $x \geq -\frac{1}{2}$ with the given options, it is clear that option C, $x \geq -\frac{1}{2}$, matches our result exactly. Therefore, option C is the correct domain for the function. Option A, $x \geq \frac{1}{2}$, is incorrect because it restricts the domain to values greater than or equal to positive one-half, which excludes values between $-\frac{1}{2}$ and $\frac{1}{2}$ that are valid in our domain. Option B, $x \geq -2$, is also incorrect because it includes values less than $-\frac{1}{2}$, which would result in a negative value inside the square root, making the function undefined in the real number system. Option D, $x > \frac{1}{2}$, is incorrect as well because it uses a strict inequality, excluding $-\frac{1}{2}$ itself, which is a valid point in our domain since the square root of zero is zero. Thus, a thorough understanding of the solution process and the meaning of the inequality symbols allows us to correctly identify the domain and choose the appropriate option. In this case, option C, $x \geq -\frac{1}{2}$, is the precise and accurate representation of the domain of the given function. This step reinforces the importance of careful comparison and attention to detail when selecting the correct answer from a set of options.

Conclusion

In conclusion, finding the domain of a function is a critical step in mathematical analysis, especially when dealing with functions like $y = 4\sqrt{4x + 2}$ that involve square roots. The domain represents the set of all possible input values (x-values) for which the function produces a real number output. For the function $y = 4\sqrt{4x + 2}$, we determined the domain by setting the radicand, $4x + 2$, greater than or equal to zero, resulting in the inequality $4x + 2 \geq 0$. Solving this inequality step-by-step, we found that $x \geq -\frac{1}{2}$. This means that the function is defined for all real numbers x that are greater than or equal to $-\frac{1}{2}$. This understanding is crucial not only for this specific problem but also for a broader range of mathematical applications. The domain is a fundamental concept in calculus, where it is used to analyze the behavior of functions, find limits, and determine continuity. Furthermore, the domain is essential in real-world applications where mathematical models are used to represent physical phenomena. For instance, in physics and engineering, understanding the domain can help ensure that the models produce meaningful and realistic results. The step-by-step approach we used – setting the radicand greater than or equal to zero, solving the inequality, and interpreting the result – provides a clear method for finding the domain of any square root function. This article has not only demonstrated the solution to the specific problem but also highlighted the broader significance of understanding domains in mathematics and its applications. Therefore, mastering the concept of domain is an investment in your mathematical toolkit, enabling you to tackle a wider array of problems with confidence and precision.