Domain And Range Of F(x) = √(4x - 3)

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In mathematics, functions are fundamental building blocks that describe relationships between variables. To thoroughly understand a function, we must delve into its domain and range. Domain refers to the set of all possible input values (x-values) for which the function is defined, while range represents the set of all possible output values (y-values) that the function can produce. In this article, we will meticulously examine the function f(x) = √(4x - 3), determining its domain and range through a step-by-step analytical approach. This process not only provides a comprehensive understanding of the function's behavior but also reinforces core concepts in mathematical analysis. By exploring these aspects, we gain insights into the function's applicability and limitations, which are crucial in various mathematical and real-world contexts. The function f(x) = √(4x - 3) presents an excellent opportunity to apply principles related to square root functions and linear inequalities, thereby enhancing our problem-solving skills in mathematics. Understanding the domain and range of functions like this is critical in many areas of mathematics, including calculus, algebra, and mathematical modeling. For instance, in calculus, the domain of a function is crucial when finding derivatives and integrals. In algebra, it helps in solving equations and inequalities. In mathematical modeling, it ensures that the model is realistic and meaningful within its context. Our examination will begin with determining the valid inputs for the function, which leads us to the concept of the domain. We will then explore the possible outputs, which will help us define the range. This comprehensive analysis will not only solidify our understanding of this particular function but also enhance our general problem-solving abilities in mathematics.

a. Determining the Domain of f(x) = √(4x - 3)

The domain of a function is the set of all possible input values (x-values) for which the function produces a real output. For the function f(x) = √(4x - 3), the domain is restricted by the square root. Specifically, the expression inside the square root (the radicand) must be greater than or equal to zero because the square root of a negative number is not a real number. Therefore, to find the domain, we need to solve the inequality:

4x−3≥04x - 3 ≥ 0

To solve this inequality, we follow these steps:

  1. Add 3 to both sides of the inequality:

    4x≥34x ≥ 3

  2. Divide both sides by 4:

    x≥34x ≥ \frac{3}{4}

This inequality tells us that the function f(x) = √(4x - 3) is defined for all real numbers x that are greater than or equal to 3/4. In interval notation, this is represented as:

[34,∞)\left[\frac{3}{4}, ∞\right)

Thus, the domain of the function f(x) = √(4x - 3) is all real numbers greater than or equal to 3/4. This restriction is crucial because it defines the set of values for which the function provides meaningful and real outputs. In practical terms, if we were to graph this function, the graph would only exist for x-values in this domain. Understanding the domain is essential for various mathematical operations, including graphing, solving equations, and analyzing the behavior of the function. For example, when solving equations involving f(x), we must ensure that any solutions lie within this domain. Furthermore, the domain plays a critical role in calculus when dealing with derivatives and integrals, as these operations are only valid within the function's domain. By rigorously determining the domain, we establish a foundational understanding of the function's applicability and limitations. This understanding ensures that we can use the function correctly and interpret its results accurately in a variety of contexts. The process of finding the domain involves not only algebraic manipulation but also a deep understanding of the nature of square root functions and their inherent constraints. This ensures that the function produces real number outputs, a fundamental requirement in many mathematical and practical applications.

b. Determining the Range of f(x) = √(4x - 3)

The range of a function is the set of all possible output values (y-values) that the function can produce. For the function f(x) = √(4x - 3), we need to consider the possible values of the square root expression. Since the square root of a non-negative number is always non-negative, the smallest value that √(4x - 3) can take is 0. This occurs when the radicand (4x - 3) is equal to 0.

As we determined in part (a), the domain of the function is x ≥ 3/4. When x = 3/4, the function value is:

f(34)=4(34)−3=3−3=0=0f\left(\frac{3}{4}\right) = \sqrt{4\left(\frac{3}{4}\right) - 3} = \sqrt{3 - 3} = \sqrt{0} = 0

As x increases beyond 3/4, the value of 4x - 3 also increases, and consequently, the value of √(4x - 3) increases. There is no upper bound on how large x can be (since the domain extends to infinity), so there is no upper bound on the values of √(4x - 3). This means that the function can take on any non-negative value.

Therefore, the range of the function f(x) = √(4x - 3) is all non-negative real numbers. In interval notation, this is represented as:

[0,∞)[0, ∞)

The range of a function is as critical as its domain in fully understanding the function's behavior. It specifies the possible output values, which helps in visualizing the function's graph and interpreting its results in various applications. For the function f(x) = √(4x - 3), the range being [0, ∞) implies that the function's graph will only exist above the x-axis (since all y-values are non-negative) and that it can achieve any non-negative y-value. This information is invaluable in fields like physics and engineering, where such functions might model real-world phenomena. For instance, the function could represent the speed of an object or the potential energy of a system, both of which cannot be negative. Moreover, understanding the range is essential when working with inverse functions. The range of a function becomes the domain of its inverse, and vice versa. Therefore, knowing the range of f(x) = √(4x - 3) is crucial if we were to find and analyze its inverse function. By determining the range, we gain a complete picture of the function's output capabilities, allowing for accurate predictions and interpretations in various mathematical and practical scenarios. This comprehensive analysis of both the domain and the range enhances our mathematical toolkit, enabling us to tackle complex problems with greater confidence and precision. The range, in conjunction with the domain, provides a complete characterization of the function, ensuring that we can apply it appropriately and interpret its results accurately in any given context.

Conclusion

In summary, for the function f(x) = √(4x - 3), we have determined that:

  • The domain is [3/4, ∞), meaning the function is defined for all real numbers greater than or equal to 3/4.
  • The range is [0, ∞), meaning the function can produce any non-negative real number.

This analysis provides a thorough understanding of the function's behavior and its limitations, which is crucial for various mathematical applications. Understanding the domain and range of a function is fundamental in mathematics as it provides a clear picture of the function's input and output capabilities. This knowledge is invaluable for graphing functions, solving equations, and applying functions in real-world contexts. For the function f(x) = √(4x - 3), determining the domain and range not only solidifies our understanding of this specific function but also reinforces key concepts applicable to a wide range of mathematical problems. The domain, being [3/4, ∞), restricts the input values to those greater than or equal to 3/4, ensuring that the radicand (4x - 3) remains non-negative. This constraint is inherent to square root functions and is crucial for obtaining real number outputs. The range, [0, ∞), indicates that the function will only produce non-negative output values, a characteristic feature of the principal square root function. Together, the domain and range offer a comprehensive view of the function's behavior. They dictate the portion of the coordinate plane where the function's graph will exist and the possible values the function can take. This understanding is paramount in advanced mathematical studies, including calculus, where the domain and range are essential for determining continuity, differentiability, and integrability. Furthermore, in practical applications, such as physics and engineering, the domain and range help ensure that mathematical models are realistic and meaningful. For instance, if this function were to model a physical quantity, the domain would represent the valid input conditions, and the range would represent the possible outcomes. Therefore, by meticulously determining and interpreting the domain and range, we equip ourselves with a powerful toolset for mathematical analysis and problem-solving.