Inverse Function Of F(x) = √(6x + 30) And Its Domain

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In the realm of mathematics, understanding the concept of inverse functions is crucial, especially when dealing with functions that have restricted domains. In this article, we will delve into the process of finding the inverse of the function f(x) = √(6x + 30), defined for the domain [-5, ∞). Furthermore, we will explore the domain of this inverse function. This exploration will not only solidify your understanding of inverse functions but also enhance your ability to manipulate functions and their domains effectively.

H2: Understanding the Function f(x) = √(6x + 30)

Before we embark on the journey of finding the inverse, let's first gain a comprehensive understanding of the function f(x) = √(6x + 30). This function is a square root function, a type of function that involves the square root of an expression. The expression inside the square root, in this case, is 6x + 30. The domain of a square root function is restricted to values that make the expression inside the square root non-negative, as the square root of a negative number is not a real number. In this specific instance, the domain is given as [-5, ∞), which ensures that 6x + 30 ≥ 0. This is because when x is -5, 6x + 30 equals 0, and for any value greater than -5, 6x + 30 will be positive. Understanding the domain of the original function is paramount as it directly influences the range of its inverse. The range of the original function, f(x) = √(6x + 30), can be determined by considering the behavior of the square root function. Since the square root of a non-negative number is always non-negative, and the smallest value inside the square root (when x = -5) is 0, the smallest value of the function is √0 = 0. As x increases, 6x + 30 also increases, and consequently, √(6x + 30) increases without bound. Therefore, the range of f(x) is [0, ∞). This information will be crucial later when we determine the domain of the inverse function. The function's behavior is also important to note; as x increases within its domain, the function f(x) also increases, indicating that it is a monotonically increasing function. This property is crucial for the existence of an inverse function, as monotonically increasing (or decreasing) functions are guaranteed to have inverses. This monotonic nature simplifies the process of finding and understanding the inverse. Additionally, the function is continuous over its domain, which is another important characteristic that contributes to the existence and properties of its inverse. Continuity ensures that there are no abrupt jumps or breaks in the function's graph, making the inverse function well-behaved. Thus, by thoroughly analyzing the original function, we lay a solid foundation for finding and interpreting its inverse.

H2: The Process of Finding the Inverse Function

The procedure for finding the inverse of a function involves a series of algebraic manipulations designed to interchange the roles of x and y. Finding the inverse function requires a systematic approach to ensure accuracy. Given the function f(x) = √(6x + 30), we begin by replacing f(x) with y, which gives us the equation y = √(6x + 30). This substitution simplifies the algebraic manipulation process. The next step is to interchange x and y, resulting in x = √(6y + 30). This interchange is the core of the inverse function process, as it effectively swaps the input and output variables. Now, our goal is to isolate y on one side of the equation. To achieve this, we first square both sides of the equation to eliminate the square root. Squaring both sides of x = √(6y + 30) yields x² = 6y + 30. This step is crucial as it transforms the equation into a form that is easier to manipulate algebraically. Next, we subtract 30 from both sides to isolate the term involving y. Subtracting 30 from both sides gives us x² - 30 = 6y. Finally, we divide both sides by 6 to solve for y. Dividing both sides by 6 results in y = (x² - 30) / 6. This equation represents the inverse function. We denote the inverse function as f⁻¹(x), so we have f⁻¹(x) = (x² - 30) / 6. This is the algebraic expression for the inverse of the original function. However, we must also consider the domain of this inverse function, which will be determined by the range of the original function. The algebraic manipulation is only half the battle; understanding the domain is equally important. The algebraic process, while straightforward, requires careful attention to avoid errors. Each step must be performed accurately to arrive at the correct inverse function. Precision in algebraic manipulation is key to success in finding inverse functions. Moreover, it’s essential to double-check each step to ensure no mistakes were made, as even a small error can lead to an incorrect inverse function. Therefore, a methodical and meticulous approach is essential when finding the inverse of a function.

H2: Determining the Domain of the Inverse Function

The domain of the inverse function, f⁻¹(x), is intrinsically linked to the range of the original function, f(x). The domain of the inverse function is equivalent to the range of the original function. This principle is a cornerstone of understanding inverse functions and their properties. As we previously established, the range of f(x) = √(6x + 30) is [0, ∞). This means that the domain of f⁻¹(x) = (x² - 30) / 6 is also [0, ∞). This restriction is crucial because the inverse function f⁻¹(x) = (x² - 30) / 6 is a quadratic function, which by itself would have a domain of all real numbers. However, because it is the inverse of a square root function, its domain is limited by the range of the original function. Understanding this relationship between range and domain is vital for defining inverse functions correctly. To further illustrate this, consider what would happen if we allowed x to be negative in the inverse function. If x were negative, there would be no corresponding y value in the original function, as the square root function only produces non-negative values. Therefore, restricting the domain of the inverse function to [0, ∞) ensures that the inverse function is properly defined and consistent with the original function. **The domain restriction ensures that the inverse function