Finding The Inverse Of F(x) = (3-x)/5 A Step-by-Step Guide

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In mathematics, inverse functions play a crucial role in reversing the operation of a given function. The inverse of a function, denoted as f-1(x), essentially undoes what the original function f(x) does. To understand this concept thoroughly, let's explore the process of finding the inverse of a function with a focus on the specific example: f(x) = (3-x)/5. This detailed exploration will help clarify the steps involved and solidify your understanding of inverse functions.

What is an Inverse Function?

Before diving into the solution, it’s important to define what an inverse function truly represents. Given a function f(x), its inverse, denoted as f-1(x), satisfies the condition that f-1(f(x)) = x and f(f-1(x)) = x. In simpler terms, if you apply a function and then apply its inverse, you'll end up with the original input. This fundamental property is the cornerstone of inverse function theory and is essential for solving related problems. For instance, consider f(x) as a machine that takes an input, processes it, and produces an output. The inverse function, f-1(x), is like a machine that takes the output and reverts it back to the original input. Understanding this reversal process is key to grasping the concept of inverse functions.

Steps to Find the Inverse

To find the inverse of a function f(x), we typically follow a series of steps. These steps ensure that we correctly reverse the operations performed by the original function. The process involves algebraic manipulation and a clear understanding of function notation. By following these steps methodically, we can accurately determine the inverse function. Let's outline these steps:

  1. Replace f(x) with y: This is a notational change to simplify the algebraic manipulation. Instead of working with f(x), we use y, which is a more common variable for representing the output of a function. This substitution makes the subsequent steps clearer and easier to follow.
  2. Swap x and y: This is the crucial step where we begin to reverse the roles of input and output. By interchanging x and y, we are setting up the equation to solve for the inverse function. This swap reflects the fundamental idea that the inverse function takes the output of the original function as its input and produces the original input as its output.
  3. Solve for y: After swapping x and y, we need to isolate y on one side of the equation. This involves using algebraic techniques such as addition, subtraction, multiplication, and division to get y by itself. The goal is to express y in terms of x, which will give us the equation for the inverse function.
  4. Replace y with f-1(x): Once we have solved for y, we replace it with the notation f-1(x). This is the final step in expressing the inverse function in standard notation. The notation f-1(x) clearly indicates that this function is the inverse of the original function f(x).

Applying the Steps to f(x) = (3-x)/5

Now, let's apply these steps to find the inverse of the given function, f(x) = (3-x)/5. This process will provide a concrete example of how to use the steps outlined above. By working through this example, you can see each step in action and understand how they contribute to finding the inverse function.

  1. Replace f(x) with y: y = (3-x)/5

    This substitution simplifies the notation and prepares the equation for the next step.

  2. Swap x and y: x = (3-y)/5

    This swap is the key to finding the inverse. We are now setting up the equation to solve for y in terms of x.

  3. Solve for y: To solve for y, we first multiply both sides by 5: 5x = 3 - y Next, we isolate y by adding y to both sides and subtracting 5x from both sides: y = 3 - 5x

    This step involves algebraic manipulation to get y by itself. We have now expressed y in terms of x.

  4. Replace y with f-1(x): f-1(x) = 3 - 5x

    This final step expresses the inverse function in standard notation. We have successfully found the inverse of f(x) = (3-x)/5.

Verifying the Inverse Function

To ensure that we have found the correct inverse function, it is crucial to verify that f-1(f(x)) = x and f(f-1(x)) = x. This verification process confirms that the inverse function truly undoes the original function. Let's verify our result for f-1(x) = 3 - 5x.

Verifying f-1(f(x)) = x

We need to substitute f(x) into f-1(x):

f-1(f(x)) = 3 - 5((3-x)/5)

Simplifying the expression:

f-1(f(x)) = 3 - (3 - x) f-1(f(x)) = 3 - 3 + x f-1(f(x)) = x

Verifying f(f-1(x)) = x

Now, we need to substitute f-1(x) into f(x):

f(f-1(x)) = (3 - (3 - 5x)) / 5

Simplifying the expression:

f(f-1(x)) = (3 - 3 + 5x) / 5 f(f-1(x)) = (5x) / 5 f(f-1(x)) = x

Since both conditions are satisfied, we can confidently conclude that f-1(x) = 3 - 5x is indeed the correct inverse function.

Conclusion

In conclusion, the inverse of the function f(x) = (3-x)/5 is f-1(x) = 3 - 5x. This result is obtained by following the standard steps for finding inverse functions: replacing f(x) with y, swapping x and y, solving for y, and replacing y with f-1(x). Additionally, we verified our result by confirming that f-1(f(x)) = x and f(f-1(x)) = x. Understanding the process of finding inverse functions is essential in mathematics, and this detailed explanation provides a solid foundation for tackling more complex problems in the future. By grasping the underlying principles and practicing these steps, you can confidently solve a wide range of inverse function problems.

Given the options:

A. f-1(x) = 3 - 5x B. f-1(x) = (5 + x) / 3 C. f-1(x) = 3 - x/5 D. f-1(x) = 5x - 3

Based on our detailed solution, the correct answer is:

A. f-1(x) = 3 - 5x

This option matches the inverse function we derived and verified through the steps outlined above. Understanding and applying the process of finding inverse functions is key to correctly identifying the answer in such problems.