Finding The Inverse Function Of H(x) A Step By Step Guide

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Introduction: Understanding Inverse Functions

In mathematics, the concept of an inverse function is fundamental. It essentially reverses the operation of a function. To grasp this, let's first define what a function is. A function, in simple terms, is a rule that assigns each input value to a unique output value. We often represent this as f(x) = y, where x is the input and y is the output. The inverse function, denoted as f⁻¹(x), does the opposite; it takes the output y and returns the original input x. Think of it as a mathematical undo button. For an inverse function to exist, the original function must be one-to-one, meaning each input corresponds to a unique output, and each output corresponds to a unique input. This ensures that the reversal process is unambiguous. Understanding the properties and how to determine the inverse function are vital in various mathematical applications, from solving equations to understanding transformations. When we look at a set of ordered pairs, as in the case of h(x) given in the problem, finding the inverse involves simply swapping the input and output values in each pair. This process highlights the fundamental principle of reversing the function's mapping, providing a clear and concise method for determining the inverse when dealing with discrete data points. Therefore, a solid understanding of functions and their inverses lays the groundwork for more advanced mathematical concepts and problem-solving techniques. Moreover, this concept extends beyond pure mathematics, finding applications in computer science, engineering, and other fields where reversing processes or mappings is crucial.

Problem Statement: Finding h⁻¹(x)

In this problem, we are given the function h(x) defined as a set of ordered pairs: h(x) = {(3,-5), (5,-7), (6,-9), (10,-12), (12,-16)}. Our task is to find the inverse function h⁻¹(x). To reiterate, the inverse function essentially reverses the roles of input and output. If a function h(x) maps x to y, then its inverse h⁻¹(x) maps y back to x. In the context of ordered pairs, this means swapping the x-coordinate and the y-coordinate in each pair. This process is straightforward but crucial for understanding how inverse functions operate when presented as discrete data points. The ordered pairs represent specific input-output relationships defined by the function h(x). To find the inverse, we need to systematically reverse these relationships. This involves identifying each pair (a, b) in h(x) and creating a new pair (b, a) for h⁻¹(x). This methodical approach ensures that we accurately capture the reversed mapping defined by the inverse function. It's a practical application of the inverse function concept, demonstrating how to translate a function's definition into its inverse counterpart. The ability to find the inverse of a function, especially when presented in this format, is a foundational skill in mathematics, and mastering this technique will help in understanding more complex mathematical operations and concepts. By carefully applying the principle of swapping coordinates, we can successfully determine the inverse function h⁻¹(x).

Solution: Determining the Inverse Function

To find the inverse function h⁻¹(x), we apply the principle of swapping the x and y coordinates in each ordered pair of the original function h(x). Given h(x) = {(3,-5), (5,-7), (6,-9), (10,-12), (12,-16)}, we proceed as follows:

  1. For the pair (3, -5), we swap the coordinates to get (-5, 3).
  2. For the pair (5, -7), we swap the coordinates to get (-7, 5).
  3. For the pair (6, -9), we swap the coordinates to get (-9, 6).
  4. For the pair (10, -12), we swap the coordinates to get (-12, 10).
  5. For the pair (12, -16), we swap the coordinates to get (-16, 12).

Therefore, the inverse function h⁻¹(x) is the set of these new ordered pairs: h⁻¹(x) = {(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)}. This process demonstrates the direct application of the definition of an inverse function to a set of discrete points. Each pair in the inverse function represents the reversal of the mapping defined by the original function. This method is particularly useful when dealing with functions defined by a finite set of points, as it provides a clear and straightforward way to determine the inverse. The resulting set of ordered pairs accurately represents the inverse function, showcasing the relationship between the original function and its inverse. Understanding this process is crucial for working with inverse functions in various contexts, including graphical representations and algebraic manipulations.

Answer and Explanation: Selecting the Correct Option

Based on our calculation, the inverse function h⁻¹(x) is given by the set of ordered pairs {(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)}. Now, let's compare this result with the given options:

  • Option A: {(3, 5), (5, 7), (6, 9), (10, 12), (12, 16)} - This option simply lists the original x and y values of h(x) with positive signs for the y values. It does not represent the inverse function.
  • **Option B: {(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)} - This option correctly swaps the x and y coordinates of each ordered pair in h(x), which aligns perfectly with our calculated inverse function.

Therefore, option B is the correct answer. This step-by-step comparison highlights the importance of accurately applying the definition of an inverse function. By carefully swapping the coordinates and comparing the result with the provided options, we can confidently identify the correct inverse function. This process reinforces the understanding of how inverse functions are derived and represented, ensuring accuracy in problem-solving. Furthermore, it demonstrates a practical approach to verifying solutions by directly comparing calculated results with available choices, which is a valuable skill in mathematical problem-solving.

Conclusion: The Significance of Inverse Functions

In conclusion, we have successfully determined the inverse function h⁻¹(x) for the given function h(x) = {(3,-5), (5,-7), (6,-9), (10,-12), (12,-16)}. By swapping the x and y coordinates in each ordered pair, we found that h⁻¹(x) = {(-5, 3), (-7, 5), (-9, 6), (-12, 10), (-16, 12)}. This exercise underscores the fundamental concept of inverse functions and their role in reversing the mapping of a function. Inverse functions are not just mathematical curiosities; they are essential tools in various areas of mathematics and its applications. They are used in solving equations, transforming graphs, and understanding relationships between different mathematical objects. The ability to find and work with inverse functions is crucial for success in higher-level mathematics and related fields. The process of finding the inverse, whether through swapping coordinates in a set of ordered pairs or using algebraic manipulations for continuous functions, reinforces the understanding of function behavior and relationships. Moreover, the concept of inverse functions extends beyond mathematics, finding applications in computer science, cryptography, and engineering, where reversing processes or mappings is a common requirement. Therefore, mastering inverse functions provides a solid foundation for both theoretical understanding and practical problem-solving in various domains.