Finding The Inverse Of Function F(x) = 1/3 - 1/21x

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Introduction

In mathematics, the concept of an inverse function is fundamental. Understanding how to find the inverse of a function is crucial for various mathematical applications, from solving equations to understanding transformations. This article delves into the process of finding the inverse of the function $f(x) = \frac{1}{3} - \frac{1}{21}x$, providing a step-by-step guide and a detailed explanation to ensure clarity and comprehension. The correct answer is B. $f^{-1}(x)=7-21 x$. We will explore the underlying principles and methodologies to solve this problem, making it an invaluable resource for students, educators, and anyone interested in mathematical problem-solving.

Understanding Inverse Functions

Before we dive into the specifics of finding the inverse of the given function, it's essential to understand what an inverse function is. In simple terms, an inverse function "undoes" what the original function does. If a function $f(x)$ takes an input $x$ and produces an output $y$, then its inverse function, denoted as $f^{-1}(x)$, takes $y$ as an input and returns the original $x$. This relationship can be mathematically expressed as:

f−1(f(x))=xf^{-1}(f(x)) = x

f(f−1(x))=xf(f^{-1}(x)) = x

For a function to have an inverse, it must be one-to-one, meaning that each input value corresponds to a unique output value, and vice versa. Graphically, a one-to-one function passes the horizontal line test, which states that no horizontal line intersects the graph of the function more than once. The process of finding an inverse function involves several key steps, which we will explore in detail as we solve the problem at hand.

Step-by-Step Process for Finding the Inverse

To find the inverse of a function, we follow a systematic approach that typically involves three main steps:

  1. Replace $f(x)$ with $y$. This step simply rewrites the function in a more convenient form for manipulation.
  2. Swap $x$ and $y$. This is the crucial step where we set up the equation to solve for the inverse function.
  3. Solve for $y$. This step involves isolating $y$ on one side of the equation, which gives us the inverse function $f^{-1}(x)$.

Understanding these steps is vital for tackling any inverse function problem. Now, let's apply these steps to the given function $f(x) = \frac{1}{3} - \frac{1}{21}x$ to find its inverse.

Applying the Steps to Find $f^{-1}(x)$

Let's walk through the process of finding the inverse of the function $f(x) = \frac{1}{3} - \frac{1}{21}x$ step-by-step. By carefully following each stage, we can arrive at the correct solution and understand the underlying methodology.

Step 1: Replace $f(x)$ with $y$

The first step in finding the inverse function is to replace $f(x)$ with $y$. This substitution makes the equation easier to manipulate in the subsequent steps. So, we rewrite the function as:

y=13−121xy = \frac{1}{3} - \frac{1}{21}x

This simple substitution sets the stage for the next crucial step, where we swap the variables $x$ and $y$.

Step 2: Swap $x$ and $y$

Now, we swap the variables $x$ and $y$ in the equation. This is a critical step because it reflects the fundamental concept of an inverse function: reversing the roles of input and output. Swapping $x$ and $y$ gives us:

x=13−121yx = \frac{1}{3} - \frac{1}{21}y

This equation now represents the inverse relationship. Our next task is to solve this equation for $y$ to express the inverse function in the standard form $f^{-1}(x) = ...$

Step 3: Solve for $y$

Solving for $y$ involves isolating $y$ on one side of the equation. To do this, we will perform a series of algebraic manipulations. First, let's subtract $\frac{1}{3}$ from both sides of the equation:

x−13=−121yx - \frac{1}{3} = -\frac{1}{21}y

Next, to isolate $y$, we need to multiply both sides of the equation by $-21$. This will cancel out the coefficient of $y$:

−21(x−13)=−21(−121y)-21(x - \frac{1}{3}) = -21(-\frac{1}{21}y)

Distributing the $-21$ on the left side, we get:

−21x+7=y-21x + 7 = y

Thus, we have solved for $y$, and the inverse function is:

f−1(x)=−21x+7f^{-1}(x) = -21x + 7

Or, rearranging the terms, we can write it as:

f−1(x)=7−21xf^{-1}(x) = 7 - 21x

This is the inverse function of the given function $f(x) = \frac{1}{3} - \frac{1}{21}x$. The correct answer is B. $f^{-1}(x) = 7 - 21x$. This result confirms that our step-by-step process has led us to the right solution.

Verification of the Inverse Function

To ensure that we have found the correct inverse function, we can verify our result by checking if $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$. This verification process is a crucial step in confirming the accuracy of our solution.

Verifying $f^{-1}(f(x)) = x$

Let's start by verifying $f^{-1}(f(x)) = x$. We substitute $f(x)$ into $f^{-1}(x)$:

f−1(f(x))=f−1(13−121x)f^{-1}(f(x)) = f^{-1}(\frac{1}{3} - \frac{1}{21}x)

Now, we replace the argument in $f^{-1}(x)$ with $\frac{1}{3} - \frac{1}{21}x$:

f−1(f(x))=7−21(13−121x)f^{-1}(f(x)) = 7 - 21(\frac{1}{3} - \frac{1}{21}x)

Distribute the $-21$:

f−1(f(x))=7−21(13)+21(121x)f^{-1}(f(x)) = 7 - 21(\frac{1}{3}) + 21(\frac{1}{21}x)

f−1(f(x))=7−7+xf^{-1}(f(x)) = 7 - 7 + x

f−1(f(x))=xf^{-1}(f(x)) = x

This confirms that $f^{-1}(f(x)) = x$, which is the first part of our verification.

Verifying $f(f^{-1}(x)) = x$

Next, we verify $f(f^{-1}(x)) = x$. We substitute $f^{-1}(x)$ into $f(x)$:

f(f−1(x))=f(7−21x)f(f^{-1}(x)) = f(7 - 21x)

Now, we replace the argument in $f(x)$ with $7 - 21x$:

f(f−1(x))=13−121(7−21x)f(f^{-1}(x)) = \frac{1}{3} - \frac{1}{21}(7 - 21x)

Distribute the $-\frac{1}{21}$:

f(f−1(x))=13−121(7)+121(21x)f(f^{-1}(x)) = \frac{1}{3} - \frac{1}{21}(7) + \frac{1}{21}(21x)

f(f−1(x))=13−13+xf(f^{-1}(x)) = \frac{1}{3} - \frac{1}{3} + x

f(f−1(x))=xf(f^{-1}(x)) = x

This confirms that $f(f^{-1}(x)) = x$, which is the second part of our verification. Since both conditions are satisfied, we can confidently say that $f^{-1}(x) = 7 - 21x$ is indeed the inverse of $f(x) = \frac{1}{3} - \frac{1}{21}x$. This verification process underscores the importance of checking our work to ensure accuracy in mathematical problem-solving.

Common Mistakes and How to Avoid Them

Finding the inverse of a function can sometimes be tricky, and there are common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure that you arrive at the correct solution. Let's discuss some of these common mistakes and strategies to avoid them.

Mistake 1: Forgetting to Swap $x$ and $y$

One of the most common mistakes is forgetting to swap the variables $x$ and $y$. This step is crucial because it sets up the equation to solve for the inverse function. Without swapping, you will not be finding the inverse but rather manipulating the original function. To avoid this mistake, always make sure that swapping $x$ and $y$ is a distinct step in your process.

Mistake 2: Incorrectly Solving for $y$

Another frequent mistake is making errors while solving for $y$. This can involve incorrect algebraic manipulations, such as mishandling fractions or signs. To avoid this mistake, take your time and carefully follow each step. Double-check your work, especially when dealing with fractions and negative signs. It can also be helpful to write out each step explicitly to minimize errors.

Mistake 3: Not Verifying the Inverse

Failing to verify the inverse is another common oversight. Verifying that $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$ is essential to confirm the accuracy of your solution. To avoid this mistake, always include verification as a final step in your process. This step can catch errors that you might have missed during the solving process.

Mistake 4: Confusing Inverse with Reciprocal

Some students confuse the inverse of a function with its reciprocal. The inverse function "undoes" the original function, while the reciprocal is simply $1$ divided by the function. These are distinct concepts, and it's crucial to understand the difference. To avoid this mistake, remember the definition of an inverse function and the steps involved in finding it.

Mistake 5: Not Recognizing Non-Invertible Functions

Not all functions have an inverse. For a function to have an inverse, it must be one-to-one. If a function is not one-to-one, it does not have an inverse. To avoid this mistake, check if the function passes the horizontal line test. If a horizontal line intersects the graph of the function more than once, the function is not one-to-one and does not have an inverse.

By being aware of these common mistakes and actively working to avoid them, you can improve your accuracy and confidence in finding inverse functions. Let’s proceed to summarize the key concepts and steps discussed in this article.

Conclusion

In this article, we have explored the process of finding the inverse of the function $f(x) = \frac{1}{3} - \frac{1}{21}x$. We began by understanding the fundamental concept of inverse functions and the steps involved in finding them. These steps include replacing $f(x)$ with $y$, swapping $x$ and $y$, and solving for $y$. By carefully applying these steps, we determined that the inverse function is $f^{-1}(x) = 7 - 21x$, which corresponds to option B.

We also emphasized the importance of verifying the inverse function by checking if $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$. This verification process ensures the accuracy of our solution and provides confidence in our result. Additionally, we discussed common mistakes that students often make when finding inverse functions and strategies to avoid them. These mistakes include forgetting to swap $x$ and $y$, incorrectly solving for $y$, not verifying the inverse, confusing inverse with reciprocal, and not recognizing non-invertible functions. By being mindful of these potential pitfalls, you can enhance your problem-solving skills and achieve greater success in mathematics.

Understanding inverse functions is a critical aspect of mathematics with wide-ranging applications in various fields. By mastering the techniques and concepts discussed in this article, you will be well-equipped to tackle more complex mathematical problems and deepen your understanding of mathematical relationships. Remember, practice and careful attention to detail are key to success in mathematics.