Finding The Inverse Of F(x) = √(3x + 2) A Step-by-Step Solution

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In mathematics, the concept of an inverse function is crucial for understanding the relationship between functions and their reversed counterparts. Given a function f(x), its inverse, denoted as f⁻¹(x), essentially undoes the operation performed by f(x). This article delves into the process of finding the inverse of a specific square root function, providing a comprehensive, step-by-step explanation suitable for students and anyone interested in deepening their understanding of inverse functions. Specifically, we will address the question: If f(x) = √(3x + 2), what is the equation for f⁻¹(x)?

Understanding Inverse Functions

Before we dive into the specifics of finding the inverse of the given square root function, it's essential to grasp the fundamental concept of inverse functions. A function can be visualized as a machine that takes an input, processes it according to a specific rule, and produces an output. The inverse function, in turn, is a machine that takes the output of the original function as its input and reverses the process to produce the original input. Mathematically, if f(a) = b, then f⁻¹(b) = a. This highlights the symmetrical relationship between a function and its inverse. Not all functions have inverses. For a function to have an inverse, it must be one-to-one, also known as injective. This means that each input value maps to a unique output value. Graphically, a one-to-one function passes the horizontal line test, which states that no horizontal line intersects the graph of the function more than once. Square root functions, under certain conditions, can be one-to-one and thus have inverses. Understanding these basic principles will greatly aid in the process of finding the inverse of f(x) = √(3x + 2). We must also consider the domain and range of the original function and its inverse, as they play a vital role in accurately defining the inverse function. The domain of f(x) becomes the range of f⁻¹(x), and the range of f(x) becomes the domain of f⁻¹(x). This reciprocal relationship is a cornerstone of inverse function theory and is crucial for ensuring the inverse function is properly defined. Furthermore, the composition of a function and its inverse results in the identity function, meaning f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This property serves as a powerful tool for verifying whether a derived inverse function is correct. In the context of our problem, we will use these principles to systematically find the inverse of the square root function and confirm its validity.

Step-by-Step Solution: Finding the Inverse of f(x) = √(3x + 2)

To determine the inverse function, f⁻¹(x), for the given function f(x) = √(3x + 2), we follow a series of algebraic steps. This process ensures we accurately reverse the operations performed by the original function. Each step is crucial in isolating x and expressing it in terms of y, which will then allow us to write the inverse function. Let's break down the process into manageable steps:

Step 1: Replace f(x) with y. This substitution is a notational convenience that simplifies the algebraic manipulations. We rewrite the function as y = √(3x + 2). This step doesn't change the function itself, but it makes it easier to work with in the subsequent steps. By using y, we can clearly see the relationship between the input (x) and the output (y), which is essential for finding the inverse. The equation y = √(3x + 2) now represents the same function as f(x) = √(3x + 2), but in a form that is more conducive to solving for x.

Step 2: Swap x and y. This is the core of the inverse function process. By interchanging x and y, we are essentially reversing the roles of the input and output. This means we now have x = √(3y + 2). This equation represents the inverse relationship, but it's not yet in the standard form of f⁻¹(x) = .... The act of swapping x and y reflects the fundamental concept of an inverse function – reversing the mapping of the original function. The new equation, x = √(3y + 2), holds the key to expressing y (which will become f⁻¹(x)) in terms of x. This is a critical step that sets the stage for solving for y and obtaining the explicit form of the inverse function.

Step 3: Solve for y. This is the algebraic manipulation phase where we isolate y on one side of the equation. Starting with x = √(3y + 2), we first square both sides to eliminate the square root, resulting in x² = 3y + 2. Squaring both sides is a valid operation as long as we consider the domain implications later. Next, we subtract 2 from both sides to isolate the term with y, giving us x² - 2 = 3y. Finally, we divide both sides by 3 to solve for y, which yields y = (x² - 2) / 3. This equation expresses y in terms of x, which is the goal of finding the inverse. However, we must remember that we squared both sides earlier, which might introduce extraneous solutions or affect the domain. Therefore, we need to consider the original function's domain and range to ensure our inverse is properly defined. The steps involved in solving for y are crucial for obtaining the explicit form of the inverse function, and each step must be performed carefully to avoid algebraic errors. The resulting equation, y = (x² - 2) / 3, is a significant milestone in our process.

Step 4: Replace y with f⁻¹(x). After isolating y, we replace it with the inverse function notation, f⁻¹(x). This gives us f⁻¹(x) = (x² - 2) / 3. This step formally expresses the inverse function in standard notation. However, we are not quite done yet. We need to consider the domain restriction. The original function f(x) = √(3x + 2) has a domain restriction because the expression under the square root must be non-negative. Specifically, 3x + 2 ≥ 0, which means x ≥ -2/3. The range of the original function is y ≥ 0, as the square root function always returns non-negative values. This range becomes the domain of the inverse function. Therefore, we must restrict the domain of f⁻¹(x) to x ≥ 0. This domain restriction is crucial for ensuring that f⁻¹(x) is a true inverse of f(x). The final form of the inverse function, including the domain restriction, is f⁻¹(x) = (x² - 2) / 3, x ≥ 0. This complete solution provides not only the equation for the inverse function but also its valid domain, which is essential for a complete mathematical definition.

Domain and Range Considerations

As mentioned earlier, understanding the domain and range of both the original function and its inverse is paramount. For the original function, f(x) = √(3x + 2), the expression inside the square root must be non-negative. This leads to the inequality 3x + 2 ≥ 0, which simplifies to x ≥ -2/3. Therefore, the domain of f(x) is [-2/3, ∞). The range of f(x) is all non-negative real numbers, since the square root function only produces non-negative outputs. Thus, the range of f(x) is [0, ∞). When we find the inverse, the domain and range swap roles. The range of f(x) becomes the domain of f⁻¹(x), and the domain of f(x) becomes the range of f⁻¹(x). This means the domain of f⁻¹(x) is [0, ∞), and the range of f⁻¹(x) is [-2/3, ∞). The domain restriction on f⁻¹(x) is crucial because the equation f⁻¹(x) = (x² - 2) / 3 is defined for all real numbers, but only the portion where x ≥ 0 corresponds to the inverse of the original square root function. If we didn't restrict the domain, f⁻¹(x) would not be a true inverse of f(x), as it would include values that do not reverse the operation of f(x). The domain and range considerations are not just a technicality; they are fundamental to the concept of inverse functions. They ensure that the inverse function