Finding The Inverse Of F(x)=(2x+1)/(-3x+5) A Step-by-Step Guide

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In mathematics, finding the inverse of a function is a fundamental concept with wide-ranging applications. The inverse function, denoted as f-1(x), essentially reverses the operation of the original function f(x). In simpler terms, if f(a) = b, then f-1(b) = a. This article will delve into the process of finding the inverse of a function, using the example f(x) = (2x + 1) / (-3x + 5) as a practical illustration. We will break down each step, providing clear explanations and insights to ensure a comprehensive understanding.

Understanding Inverse Functions

Before diving into the mechanics of finding the inverse, it’s crucial to grasp the underlying concept. Think of a function as a machine that takes an input, processes it, and produces an output. The inverse function is like a machine that reverses this process, taking the output and returning the original input. However, 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 output corresponds to a unique input. Graphically, a one-to-one function passes the horizontal line test, which states that no horizontal line intersects the graph more than once.

The concept of inverse functions is deeply intertwined with the idea of function composition. If we compose a function with its inverse, the result is the identity function, x. That is, f-1(f(x)) = x and f(f-1(x)) = x. This property serves as a powerful tool for verifying whether a function we've found is indeed the inverse.

In the context of real-world applications, inverse functions are used extensively in various fields. For instance, in cryptography, inverse functions play a crucial role in encoding and decoding messages. In economics, they are used to determine the demand function given the supply function, or vice versa. Understanding inverse functions is therefore not just a theoretical exercise but a practical skill with significant real-world implications. Moreover, the ability to find inverse functions is essential in solving equations and understanding the behavior of mathematical models across various disciplines.

Step 1: Replace f(x) with y

The first step in finding the inverse of a function is to replace the function notation f(x) with the variable y. This seemingly simple step helps to clarify the relationship between the input and output variables and makes the subsequent algebraic manipulations easier to follow. So, for our example, f(x) = (2x + 1) / (-3x + 5), we rewrite it as:

y = (2x + 1) / (-3x + 5)

This substitution is a standard practice in mathematics when dealing with functions and their inverses. It allows us to treat the function as an equation and manipulate it algebraically to solve for the inverse. The variable y now represents the output of the function for a given input x. This step is crucial because it sets the stage for the next step, which involves swapping the roles of x and y.

Step 2: Swap x and y

This is the core step in finding the inverse function. We interchange the roles of x and y. This reflects the fundamental idea of an inverse function – it reverses the roles of input and output. So, in our equation, we replace every instance of y with x and every instance of x with y. This gives us:

x = (2y + 1) / (-3y + 5)

This equation now represents the inverse relationship. We have essentially transformed the original equation, which expressed y in terms of x, into an equation that expresses x in terms of y. The next step is to solve this equation for y, which will give us the inverse function in the standard notation.

Step 3: Solve for y

Now, we need to isolate y in the equation x = (2y + 1) / (-3y + 5). This involves a series of algebraic manipulations. First, we multiply both sides of the equation by (-3y + 5) to eliminate the denominator:

x(-3y + 5) = 2y + 1

Next, we distribute x on the left side:

-3xy + 5x = 2y + 1

Our goal is to get all the terms containing y on one side of the equation and all the other terms on the other side. So, we add 3xy to both sides and subtract 1 from both sides:

5x - 1 = 2y + 3xy

Now, we can factor out y from the right side:

5x - 1 = y(2 + 3x)

Finally, we divide both sides by (2 + 3x) to isolate y:

y = (5x - 1) / (3x + 2)

This equation expresses y in terms of x, which is what we need for the inverse function.

Step 4: Replace y with f-1(x)

The final step is to replace y with the inverse function notation f-1(x). This gives us the inverse function in its standard form. So, we have:

f-1(x) = (5x - 1) / (3x + 2)

This is the inverse of the original function f(x) = (2x + 1) / (-3x + 5). We have successfully found the inverse function by following the steps of replacing f(x) with y, swapping x and y, solving for y, and then replacing y with f-1(x).

Verifying the Inverse Function

To ensure that we have correctly found the inverse function, we can verify it by checking if the composition of the function and its inverse results in the identity function, x. We need to check both f-1(f(x)) = x and f(f-1(x)) = x.

Let's start with f-1(f(x)):

f-1(f(x)) = f-1((2x + 1) / (-3x + 5))

We substitute (2x + 1) / (-3x + 5) into f-1(x):

f-1(f(x)) = [5((2x + 1) / (-3x + 5)) - 1] / [3((2x + 1) / (-3x + 5)) + 2]

To simplify this expression, we multiply the numerator and denominator of the main fraction by (-3x + 5):

f-1(f(x)) = [5(2x + 1) - (-3x + 5)] / [3(2x + 1) + 2(-3x + 5)]

Expanding the terms, we get:

f-1(f(x)) = (10x + 5 + 3x - 5) / (6x + 3 - 6x + 10)

Simplifying further:

f-1(f(x)) = (13x) / 13

f-1(f(x)) = x

Now, let's check f(f-1(x)):

f(f-1(x)) = f((5x - 1) / (3x + 2))

We substitute (5x - 1) / (3x + 2) into f(x):

f(f-1(x)) = [2((5x - 1) / (3x + 2)) + 1] / [-3((5x - 1) / (3x + 2)) + 5]

Again, we multiply the numerator and denominator of the main fraction by (3x + 2):

f(f-1(x)) = [2(5x - 1) + (3x + 2)] / [-3(5x - 1) + 5(3x + 2)]

Expanding the terms, we get:

f(f-1(x)) = (10x - 2 + 3x + 2) / (-15x + 3 + 15x + 10)

Simplifying further:

f(f-1(x)) = (13x) / 13

f(f-1(x)) = x

Since both f-1(f(x)) = x and f(f-1(x)) = x, we have verified that f-1(x) = (5x - 1) / (3x + 2) is indeed the inverse of f(x) = (2x + 1) / (-3x + 5).

Conclusion

Finding the inverse of a function involves a systematic process of replacing f(x) with y, swapping x and y, solving for y, and then replacing y with f-1(x). By following these steps, we can successfully find the inverse of a given function. It's crucial to remember the importance of verifying the inverse by checking the composition of the function and its inverse. This ensures that the result is accurate. The example f(x) = (2x + 1) / (-3x + 5) illustrates this process effectively. Understanding inverse functions is not only essential in mathematics but also has significant applications in various fields, making it a valuable concept to master.