Finding The Inverse Function Of F(x) = ((7x + 3) / (x - 4)) / (x - 4)

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Introduction to Inverse Functions

In the realm of mathematics, the concept of an inverse function holds significant importance. An inverse function, denoted as f⁻¹(x), essentially reverses the operation performed by the original function, f(x). In simpler terms, if f(a) = b, then f⁻¹(b) = a. Finding the inverse of a function is a fundamental skill in various mathematical fields, including calculus, algebra, and analysis. This article delves into the process of finding the inverse function for the given function f(x) = ((7x + 3) / (x - 4)) / (x - 4). Understanding inverse functions is crucial for solving equations, analyzing function behavior, and comprehending mathematical relationships. The quest to find an inverse function is not merely an exercise in algebraic manipulation; it's a journey into understanding the fundamental nature of mathematical operations and their reversibility. Before embarking on the process of finding the inverse, it's crucial to ensure the function is one-to-one, also known as injective. A one-to-one function guarantees a unique output for each input, allowing for a well-defined inverse. If a function fails this criterion, it may be necessary to restrict its domain to create a section that is one-to-one, enabling the determination of a partial inverse. The inverse function provides a powerful tool for unraveling the original function's actions, offering a reverse pathway to navigate the mathematical landscape. The process of finding the inverse often involves a series of algebraic steps, each designed to isolate the independent variable and express it in terms of the dependent variable. This journey of algebraic manipulation highlights the interconnectedness of mathematical operations and the elegant dance of variables as we seek to reverse the function's flow.

Problem Statement: Unraveling f(x) = ((7x + 3) / (x - 4)) / (x - 4)

Our primary objective is to determine the inverse function, f⁻¹(x), for the given function f(x) = ((7x + 3) / (x - 4)) / (x - 4). This function presents a fascinating challenge due to its structure, which involves a rational expression divided by another term. To successfully find the inverse, we need to meticulously dissect the function, understand its behavior, and then reverse its operations step by step. The given function is a rational function, characterized by a polynomial divided by another polynomial. In this case, we have a fraction within a fraction, making it essential to simplify the function before attempting to find its inverse. This simplification process will involve algebraic manipulations that combine the fractions and express the function in a more manageable form. The challenge in finding the inverse of this particular function lies in the repeated division by (x - 4). This necessitates careful consideration of the domain of the function and the restrictions it imposes. We must ensure that our inverse function accurately reflects these restrictions and avoids any undefined values. The process of finding the inverse will involve swapping the roles of x and y, followed by isolating y. This algebraic dance requires a keen eye for detail and a solid understanding of algebraic principles. We will encounter steps that involve multiplying both sides by expressions containing y, and we must be vigilant about maintaining equality throughout the process. The journey to finding the inverse function for f(x) = ((7x + 3) / (x - 4)) / (x - 4) is not just about arriving at a final answer; it's about the intellectual exploration of mathematical reversibility and the elegant interplay of algebraic manipulations. Each step we take is a deliberate move in a carefully choreographed dance, guided by the principles of mathematics and the pursuit of a profound understanding of the function's inverse nature.

Step-by-Step Solution: Finding the Inverse

Let's embark on the journey of finding the inverse function, f⁻¹(x), for f(x) = ((7x + 3) / (x - 4)) / (x - 4). This will involve a series of algebraic manipulations, each carefully designed to reverse the operations performed by the original function.

  1. Replace f(x) with y: This is a standard first step in finding inverse functions, making the equation easier to manipulate. So, we rewrite the equation as:

    y = ((7x + 3) / (x - 4)) / (x - 4)

  2. Simplify the Expression: We can simplify the compound fraction by dividing (7x + 3) / (x - 4) by (x - 4), which is the same as multiplying by 1/(x - 4). This gives us:

    y = (7x + 3) / ((x - 4) * (x - 4))

    y = (7x + 3) / (x - 4)²

  3. Swap x and y: The essence of finding an inverse function lies in swapping the roles of the input (x) and the output (y). This reflects the idea that the inverse function reverses the original function's mapping. Swapping x and y, we get:

    x = (7y + 3) / (y - 4)²

  4. Isolate y: Now, the crucial step is to isolate y. This involves a series of algebraic maneuvers to get y by itself on one side of the equation.

    Multiply both sides by (y - 4)²:

    x(y - 4)² = 7y + 3

    Expand the left side:

    x(y² - 8y + 16) = 7y + 3

    xy² - 8xy + 16x = 7y + 3

  5. Rearrange into a Quadratic Equation: Our goal is to solve for y, and the presence of y² suggests a quadratic equation. To solve a quadratic, we need to get all the terms on one side, leaving zero on the other:

    xy² - 8xy - 7y + 16x - 3 = 0

    xy² + (-8x - 7)y + (16x - 3) = 0

  6. Solve the Quadratic Equation for y: We now have a quadratic equation in the form ay² + by + c = 0, where:

    a = x

    b = -8x - 7

    c = 16x - 3

    We can use the quadratic formula to solve for y:

    y = (-b ± √(b² - 4ac)) / (2a)

    Substituting the values of a, b, and c, we get:

    y = (8x + 7 ± √((-8x - 7)² - 4 * x * (16x - 3))) / (2x)

    y = (8x + 7 ± √(64x² + 112x + 49 - 64x² + 12x)) / (2x)

    y = (8x + 7 ± √(124x + 49)) / (2x)

  7. Express the Inverse Function: Finally, we replace y with f⁻¹(x) to express the inverse function:

    f⁻¹(x) = (8x + 7 ± √(124x + 49)) / (2x)

Verification and Domain Considerations

Now that we have derived the inverse function, f⁻¹(x) = (8x + 7 ± √(124x + 49)) / (2x), it's crucial to verify our result and consider the domain of both the original function and its inverse. Verification ensures that the derived inverse function indeed reverses the operations of the original function, while domain considerations are essential to identify any restrictions or limitations on the input values.

Verification: To verify the inverse function, we can use the property that f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This means that if we plug the inverse function into the original function, or vice versa, we should obtain x as the result. Let's perform one of these checks:

f(f⁻¹(x)) = f((8x + 7 ± √(124x + 49)) / (2x))

This substitution leads to a complex algebraic expression, and simplifying it directly can be quite challenging. However, the principle remains the same: if we were to fully simplify this expression, we should arrive at x. Due to the complexity, we'll rely on the algebraic correctness of our steps in the derivation process as a primary form of verification.

Domain Considerations: The domain of a function is the set of all possible input values (x-values) for which the function is defined. For rational functions, we need to consider values that would make the denominator zero, as division by zero is undefined. For functions involving square roots, we need to ensure that the expression inside the square root is non-negative.

  • Original Function, f(x) = ((7x + 3) / (x - 4)) / (x - 4) = (7x + 3) / (x - 4)²: The denominator is (x - 4)², which becomes zero when x = 4. Therefore, the domain of f(x) is all real numbers except x = 4.

  • Inverse Function, f⁻¹(x) = (8x + 7 ± √(124x + 49)) / (2x): We have two potential restrictions:

    1. The denominator, 2x, cannot be zero, so x ≠ 0.

    2. The expression inside the square root, 124x + 49, must be non-negative:

      124x + 49 ≥ 0

      124x ≥ -49

      x ≥ -49/124

    Therefore, the domain of f⁻¹(x) is x ≥ -49/124 and x ≠ 0. The range of the original function corresponds to the domain of the inverse function, and vice versa. Analyzing the range of f(x) is complex and may require calculus techniques. However, understanding the domain restrictions provides valuable insights into the behavior of both the original function and its inverse.

Conclusion: The Inverse Function Revealed

In conclusion, we have successfully determined the inverse function for f(x) = ((7x + 3) / (x - 4)) / (x - 4). Through a series of algebraic manipulations, including simplification, swapping variables, and solving a quadratic equation, we arrived at the inverse function: f⁻¹(x) = (8x + 7 ± √(124x + 49)) / (2x). This journey of finding the inverse function highlights the elegance and power of algebraic techniques in reversing mathematical operations. The process involved several key steps, each requiring careful attention to detail and a solid understanding of mathematical principles. We began by simplifying the original function, then swapped the roles of x and y, and finally, isolated y using the quadratic formula. This process showcased the interconnectedness of various mathematical concepts and the importance of a systematic approach to problem-solving. Furthermore, we emphasized the importance of verification and domain considerations. Verifying the inverse function ensures that our result is accurate and that the derived function truly reverses the original function's operations. Domain considerations are crucial for understanding the limitations and restrictions on the input values for both the original function and its inverse. The domain of f⁻¹(x) is x ≥ -49/124 and x ≠ 0, reflecting the inherent constraints imposed by the square root and the denominator in the inverse function's expression. The inverse function we found is not a simple one; it involves a plus-or-minus sign, indicating that there are potentially two branches to the inverse function. This complexity arises from the quadratic nature of the relationship between x and y in the original function. Understanding the concept of inverse functions is essential in various areas of mathematics and its applications. Inverse functions provide a powerful tool for solving equations, analyzing function behavior, and modeling real-world phenomena. The process of finding an inverse function not only enhances our algebraic skills but also deepens our understanding of mathematical relationships and the reversibility of mathematical operations.