Finding The Inverse Of F(x) = 7x + 2 A Step-by-Step Guide

In mathematics, understanding inverse functions is crucial for various applications, from solving equations to analyzing transformations. The inverse of a function, denoted as $f^{-1}(x)$, essentially "undoes" what the original function $f(x)$ does. In simpler terms, if $f(a) = b$, then $f^{-1}(b) = a$. This concept is particularly useful when dealing with linear functions, which have straightforward inverses. In this comprehensive guide, we will explore the process of finding the inverse of the linear function $f(x) = 7x + 2$, providing a step-by-step explanation to ensure clarity and understanding.

To begin, it's essential to grasp the fundamental principle behind inverse functions. A function $f(x)$ takes an input $x$ and produces an output $y$. The inverse function, $f^{-1}(x)$, reverses this process; it takes the output $y$ and returns the original input $x$. Mathematically, this can be expressed as $f^{-1}(f(x)) = x$ and $f(f^{-1}(x)) = x$. This property serves as a crucial check to verify whether a function is indeed the inverse of another. When dealing with linear functions, the inverse will also be a linear function, making the process relatively straightforward. However, it's important to follow a systematic approach to avoid common errors. The steps typically involve swapping the roles of $x$ and $y$, and then solving for $y$. This process effectively reverses the operations performed by the original function, leading us to its inverse.

Understanding the properties of linear functions is also beneficial in this context. Linear functions have a constant rate of change, represented by their slope, and a y-intercept that indicates where the function crosses the y-axis. When finding the inverse, the slope and intercept will transform in a predictable manner. For example, if the original function has a slope of $m$, the inverse function will have a slope of $\frac{1}{m}$, provided $m$ is not zero. The y-intercept will also shift, reflecting the reversed relationship between the input and output values. By keeping these properties in mind, you can develop a better intuition for what the inverse function should look like, which can help in verifying your solution.

Let's dive into the process of finding the inverse of the given linear function $f(x) = 7x + 2$. This function multiplies the input $x$ by 7 and then adds 2 to the result. To find the inverse, we need to reverse these operations in the opposite order. The following steps outline the procedure:

1. Replace $f(x)$ with $y$: This substitution makes the equation easier to manipulate. So, we rewrite $f(x) = 7x + 2$ as $y = 7x + 2$.

2. Swap $x$ and $y$: This is the key step in finding the inverse. We interchange the roles of $x$ and $y$, which gives us $x = 7y + 2$. This reflects the inverse relationship where the original output becomes the new input, and vice versa.

3. Solve for $y$: Now, we need to isolate $y$ on one side of the equation. First, subtract 2 from both sides: $x - 2 = 7y$. Next, divide both sides by 7 to get $y = \frac{x - 2}{7}$. This equation represents the inverse function.

4. Replace $y$ with $f^{-1}(x)$: To denote that this is the inverse function, we replace $y$ with $f^{-1}(x)$. Thus, the inverse of $f(x)$ is $f^{-1}(x) = \frac{x - 2}{7}$.

This step-by-step approach ensures a clear and logical progression towards finding the inverse function. By carefully reversing the operations performed by the original function, we arrive at the inverse function, which undoes the effect of the original function. It's always a good practice to verify the inverse by checking if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$.

To further illustrate this process, let's consider a numerical example. If we input $x = 1$ into the original function $f(x) = 7x + 2$, we get $f(1) = 7(1) + 2 = 9$. Now, let's input this result into the inverse function $f^{-1}(x) = \frac{x - 2}{7}$. We get $f^{-1}(9) = \frac{9 - 2}{7} = \frac{7}{7} = 1$, which is our original input. This confirms that the inverse function correctly reverses the operation of the original function.

After finding a potential inverse function, it's crucial to verify its correctness. The most reliable method for verifying an inverse function is to check if $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. This ensures that the inverse function truly "undoes" the original function.

Let's verify our inverse function $f^{-1}(x) = \frac{x - 2}{7}$ for $f(x) = 7x + 2$:

1. Check $f(f^{-1}(x))$: We need to substitute $f^{-1}(x)$ into $f(x)$.

   $$f(f^{-1}(x)) = f(\frac{x - 2}{7}) = 7(\frac{x - 2}{7}) + 2$$

   Simplifying, we get:

   $$= (x - 2) + 2 = x$$

   This confirms that $f(f^{-1}(x)) = x$.

2. Check $f^{-1}(f(x))$: Now, we substitute $f(x)$ into $f^{-1}(x)$.

   $$f^{-1}(f(x)) = f^{-1}(7x + 2) = \frac{(7x + 2) - 2}{7}$$

   Simplifying, we get:

   $$= \frac{7x}{7} = x$$

   This confirms that $f^{-1}(f(x)) = x$.

Since both conditions are satisfied, we can confidently conclude that $f^{-1}(x) = \frac{x - 2}{7}$ is indeed the correct inverse function for $f(x) = 7x + 2$. This verification step is essential, especially in more complex scenarios where the inverse might not be immediately obvious.

Verifying the inverse function also provides a deeper understanding of the relationship between a function and its inverse. It demonstrates the symmetry inherent in the concept of inverse functions, where the input and output roles are reversed. This symmetry is visually represented by the graphs of the function and its inverse, which are reflections of each other across the line $y = x$. By understanding this graphical relationship, you can often anticipate the form of the inverse function and check your algebraic manipulations against your visual intuition.

Now that we have found the inverse function, let's compare our result with the given options:

A. $f^{-1}(x) = \frac{x - 7}{2}$ B. $f^{-1}(x) = \frac{x - 2}{7}$ C. $f^{-1}(x) = 7x - 2$ D. $f^{-1}(x) = 2 - 7x$

Our calculated inverse function, $f^{-1}(x) = \frac{x - 2}{7}$, matches option B. Therefore, option B is the correct answer. The other options can be ruled out through our step-by-step solution and verification process.

To further clarify why the other options are incorrect, let's briefly analyze them. Option A, $f^{-1}(x) = \frac{x - 7}{2}$, incorrectly subtracts 7 instead of 2 and divides by 2 instead of 7. Option C, $f^{-1}(x) = 7x - 2$, does not correctly reverse the operations of the original function; it maintains the multiplication by 7 but subtracts 2 instead of dividing by 7 after subtracting 2. Option D, $f^{-1}(x) = 2 - 7x$, incorrectly reverses the order of operations and changes the signs in a way that does not align with the inverse relationship.

By systematically working through the problem and comparing our solution with the given options, we can confidently identify the correct answer. This process also reinforces the importance of understanding the underlying principles of inverse functions and the steps involved in finding them.

In conclusion, the inverse of the function $f(x) = 7x + 2$ is $f^{-1}(x) = \frac{x - 2}{7}$. This was determined by replacing $f(x)$ with $y$, swapping $x$ and $y$, solving for $y$, and then replacing $y$ with $f^{-1}(x)$. We also verified our result by confirming that $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. This detailed exploration not only provides the correct answer but also reinforces the methodology for finding and verifying inverse functions, a fundamental concept in mathematics. Understanding inverse functions is essential for various mathematical applications, and mastering the techniques to find them will undoubtedly enhance your problem-solving skills in more complex scenarios. Remember to always follow a systematic approach, and don't forget to verify your solution to ensure accuracy.