Inverse Of F(x) = 4x 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 operations. The question at hand asks us to identify the inverse of the function f(x) = 4x. This article will delve deep into the process of finding inverse functions, offering a step-by-step guide to solving this specific problem, and providing a broader understanding of inverse functions in general. Understanding the inverse of a function is not merely an academic exercise; it is a foundational concept that underpins many areas of mathematics and its applications in the real world. From solving equations to understanding transformations, inverse functions play a pivotal role. Therefore, a solid grasp of this concept is essential for anyone pursuing studies in mathematics, science, or engineering.

Understanding Inverse Functions

Before we tackle the specific problem, let's establish a clear understanding of what an inverse function is. In simple terms, an inverse function "undoes" what the original function does. If a function f takes an input x and produces an output y, then the inverse function, denoted as f⁻¹, takes y as input and returns x. Mathematically, this can be expressed as follows:

If f(x) = y, then f⁻¹(y) = x

This relationship highlights the symmetrical nature of functions and their inverses. The inverse function essentially reverses the mapping performed by the original function. To illustrate this with a simple example, consider the function f(x) = x + 2. This function adds 2 to any input x. The inverse function, therefore, should subtract 2 from any input. Indeed, the inverse function is f⁻¹(x) = x - 2. If we apply f to x and then apply f⁻¹ to the result, we get back our original input x. This is a key characteristic of inverse functions.

How to Find the Inverse Function

The process of finding an inverse function involves a few key steps. These steps are designed to systematically reverse the operations performed by the original function. Let's outline these steps:

  1. Replace f(x) with y: This is a simple notational change that makes the subsequent steps easier to follow. We are essentially rewriting the function in terms of x and y.
  2. Swap x and y: This is the crucial step in finding the inverse function. By swapping x and y, we are reflecting the function across the line y = x, which is the graphical representation of the inverse operation.
  3. Solve for y: After swapping x and y, we need to isolate y on one side of the equation. This involves using algebraic manipulations to undo the operations that were originally performed on x.
  4. Replace y with f⁻¹(x): Finally, we replace y with the notation for the inverse function, f⁻¹(x). This gives us the inverse function in standard notation.

Let's illustrate this process with an example. Consider the function f(x) = 2x + 3. To find its inverse function, we follow these steps:

  1. Replace f(x) with y: y = 2x + 3
  2. Swap x and y: x = 2y + 3
  3. Solve for y:
    • Subtract 3 from both sides: x - 3 = 2y
    • Divide both sides by 2: y = (x - 3) / 2
  4. Replace y with f⁻¹(x): f⁻¹(x) = (x - 3) / 2

Therefore, the inverse function of f(x) = 2x + 3 is f⁻¹(x) = (x - 3) / 2. This process can be applied to a wide range of functions to find their inverses.

Applying the Process to f(x) = 4x

Now, let's apply the steps outlined above to the function given in the problem, f(x) = 4x. This function represents a simple linear relationship where the output is four times the input. To find the inverse function, we follow the same steps:

  1. Replace f(x) with y: y = 4x
  2. Swap x and y: x = 4y
  3. Solve for y:
    • Divide both sides by 4: y = x / 4
  4. Replace y with f⁻¹(x): f⁻¹(x) = x / 4

Thus, the inverse function of f(x) = 4x is f⁻¹(x) = x / 4. This means that the inverse function takes an input x and divides it by 4. This is the opposite operation of the original function, which multiplied the input by 4.

Analyzing the Answer Choices

Now that we have found the inverse function, let's compare it to the answer choices provided in the problem:

A. h(x) = x + 4 B. h(x) = x - 4 C. h(x) = (1/4)ˣ D. h(x) = (1/4)x

We found that the inverse function is f⁻¹(x) = x / 4, which is equivalent to h(x) = (1/4)x. Therefore, the correct answer is D. The other options represent different types of functions:

  • Option A, h(x) = x + 4, represents a linear function that adds 4 to the input. This is not the inverse function as it performs addition instead of division.
  • Option B, h(x) = x - 4, represents a linear function that subtracts 4 from the input. This is also not the inverse function for the same reason.
  • Option C, h(x) = (1/4)ˣ, represents an exponential function. Exponential functions have inverse functions that are logarithmic functions, not linear functions like the inverse we are seeking.

Therefore, only option D correctly represents the inverse function of f(x) = 4x.

Graphical Representation of Inverse Functions

Visualizing functions and their inverses graphically can provide a deeper understanding of the concept. The graph of a function and its inverse are reflections of each other across the line y = x. This is a direct consequence of the swapping of x and y in the process of finding the inverse function.

To illustrate this, let's consider the function f(x) = 4x and its inverse f⁻¹(x) = x / 4. The graph of f(x) = 4x is a straight line passing through the origin with a slope of 4. The graph of f⁻¹(x) = x / 4 is also a straight line passing through the origin, but with a slope of 1/4. If you were to draw these two lines on the same coordinate plane, you would see that they are reflections of each other across the line y = x. This graphical relationship is a useful tool for verifying that you have correctly found the inverse function.

Importance of Understanding Inverse Functions

The concept of inverse functions is not just a theoretical exercise; it has practical applications in various fields. Understanding inverse functions is crucial for:

  • Solving Equations: Inverse functions are used to isolate variables in equations. For example, if we have the equation 4x = 8, we can use the inverse function of f(x) = 4x, which is f⁻¹(x) = x / 4, to solve for x. Applying the inverse function to both sides of the equation gives us x = 8 / 4, so x = 2.
  • Transformations: Inverse functions are used to undo transformations. For example, if we apply a transformation to a set of data, we can use the inverse function to revert the data back to its original state.
  • Cryptography: Inverse functions play a critical role in cryptography, where they are used to encrypt and decrypt messages. Encryption involves applying a function to a message to make it unreadable, and decryption involves applying the inverse function to recover the original message.
  • Calculus: Inverse functions are essential in calculus, particularly in the study of derivatives and integrals. The derivative of an inverse function is related to the derivative of the original function, and the integral of an inverse function can be calculated using techniques involving the original function.

In conclusion, understanding inverse functions is a fundamental skill in mathematics with far-reaching applications. The ability to find and manipulate inverse functions is essential for solving problems in algebra, calculus, and other areas of mathematics, as well as in various real-world applications.

Conclusion

In this article, we have explored the concept of inverse functions and demonstrated how to find the inverse of a given function. We specifically addressed the question of finding the inverse of f(x) = 4x, and through a step-by-step process, we determined that the correct answer is D. h(x) = (1/4)x. We also discussed the graphical representation of inverse functions and highlighted the importance of understanding inverse functions in various mathematical and real-world contexts. Mastering the concept of inverse functions is a valuable asset for anyone pursuing studies in mathematics, science, or engineering.