Finding The Inverse Function G(x) For F(x) = 4x + 12

Hey guys! Let's dive into a fun math problem where we need to find the inverse of a function. Understanding inverse functions is super useful, especially when you're dealing with more complex math concepts later on. So, let's break this down step by step.

Understanding Inverse Functions

Before we jump into the specific problem, let's quickly recap what inverse functions are all about. In simple terms, an inverse function undoes what the original function does. Think of it like this: if you have a function that adds 5 to a number, the inverse function would subtract 5 from that number. Mathematically, if we have a function f(x), its inverse is denoted as g(x) or f⁻¹(x). The key relationship here is that if you apply f to x and then apply g to the result, you get back x. This can be written as g(f(x)) = x and f(g(x)) = x. This concept is crucial for solving the problem at hand. Finding the inverse function involves a few key steps, which we'll walk through in detail as we solve the given problem. Remember, the goal is to isolate x in terms of y, and then swap x and y to get the inverse function in the standard notation. This process might seem a bit abstract at first, but with practice, it becomes second nature. We'll make sure to explain each step clearly so you can follow along and understand the logic behind it. Keep in mind that not every function has an inverse; for a function to have an inverse, it must be one-to-one, meaning that it passes both the vertical and horizontal line tests. However, for linear functions like the one we're dealing with today, this is generally not an issue. We can confidently proceed with finding the inverse. The concept of inverse functions extends beyond just basic algebra; it's a fundamental idea in calculus and other advanced mathematical fields. For instance, understanding inverses is essential when working with trigonometric functions and their inverses (like arcsin, arccos, and arctan). It also plays a role in solving differential equations and understanding transformations of functions. So, mastering this concept now will set you up for success in your future math endeavors.

The Problem: Finding g(x)

Okay, so here's the problem we're tackling: If g(x) is the inverse of f(x) and f(x) = 4x + 12, what is g(x)? We've got four options to choose from:

A. g(x) = 12x + 4 B. g(x) = (1/4)x - 12 C. g(x) = x - 3 D. g(x) = (1/4)x - 3

Let's break down how to find the correct answer. To find the inverse function, we need to reverse the operations that f(x) performs. f(x) multiplies x by 4 and then adds 12. So, g(x) will need to undo these operations in the reverse order. This means we'll first undo the addition of 12, and then undo the multiplication by 4. We can approach this systematically by replacing f(x) with y, then swapping x and y, and finally solving for y. This method ensures that we're correctly reversing the operations and finding the inverse function. Another way to think about it is that the inverse function essentially reflects the original function across the line y = x. This graphical interpretation can be helpful in visualizing the relationship between a function and its inverse. However, for this problem, we'll focus on the algebraic method of swapping variables and solving for y. This approach is generally more straightforward and less prone to errors, especially when dealing with more complex functions. As we work through the steps, pay close attention to the order of operations and how we're isolating y. This is a common technique used in many mathematical problems, so mastering it here will be beneficial in the long run. Remember, the goal is to express y in terms of x, which will give us the equation for the inverse function g(x). So, let's get started and see which of the options is the correct inverse of f(x) = 4x + 12. We'll go through each step meticulously to make sure we arrive at the right answer. Stay focused and let's solve this together!

Step-by-Step Solution

Here's how we can solve this step by step:

1. Replace f(x) with y: So, we have y = 4x + 12.
2. Swap x and y: This gives us x = 4y + 12. This step is crucial because it's where we actually start reversing the roles of the input and output variables. By swapping x and y, we're essentially setting up the equation to solve for the inverse function. It's like looking at the original function from the opposite perspective. Instead of finding y given x, we're now finding x given y. This swap is the heart of the inverse function process. It reflects the function across the line y = x, which is the graphical representation of finding an inverse. Without this step, we wouldn't be able to correctly reverse the operations and find the inverse. So, always remember to swap x and y as the second step in finding the inverse function. It's a simple but incredibly important step that sets the stage for solving for the inverse. Make sure you understand why we do this swap – it's not just a random step, it's the foundation of the whole process. With this swap, we've now transformed the equation into a form where we can isolate y and express it in terms of x, which is exactly what we need to do to find the inverse function.
3. Solve for y: Now, we need to isolate y in the equation x = 4y + 12. First, subtract 12 from both sides: x - 12 = 4y. Then, divide both sides by 4: (x - 12) / 4 = y. This is where we undo the operations that were performed on x in the original function, but in reverse order. We first undid the addition of 12 by subtracting it, and then we undid the multiplication by 4 by dividing. Isolating y is the key to finding the inverse function because it allows us to express the inverse function in the standard form y = g(x). Each step we take in solving for y is a crucial part of the process, and it's important to understand why we're doing what we're doing. By carefully following these steps, we can confidently find the inverse of any function, as long as it exists. Remember, the inverse function undoes the operations of the original function, and solving for y is the way we mathematically represent that undoing. So, pay close attention to the algebraic manipulations involved in isolating y, and you'll be well on your way to mastering inverse functions. This step not only gives us the equation for the inverse function, but it also reinforces the concept that the inverse function reverses the operations of the original function. With y now isolated, we're just one step away from having our answer.
4. Rewrite y as g(x): So, g(x) = (x - 12) / 4. We can simplify this further: g(x) = x/4 - 12/4 = (1/4)x - 3. This final step is crucial because it clearly shows us the equation of the inverse function. By rewriting y as g(x), we're explicitly stating that we've found the inverse function. The simplified form, g(x) = (1/4)x - 3, makes it easy to see the relationship between x and its inverse. Rewriting y as g(x) is not just a notational change; it's a way of formally declaring that we've successfully found the inverse function. This notation is universally understood in mathematics, and it's important to use it correctly. It also helps to avoid confusion between the original function and its inverse. By using the notation g(x), we clearly distinguish the inverse function from the original function f(x). This distinction is essential for clear communication and accurate problem-solving. So, always remember to rewrite y as g(x) once you've solved for it – it's the final touch that completes the process of finding the inverse function. The final equation, g(x) = (1/4)x - 3, is the answer we've been looking for, and it represents the inverse function of f(x) = 4x + 12. With this equation, we can now find the input value that corresponds to any output value of the original function, and vice versa. This is the power of inverse functions – they allow us to reverse the process and gain a deeper understanding of the relationship between the input and output variables.

The Answer

Looking at our options, the correct answer is D. g(x) = (1/4)x - 3. We did it, guys!

Key Takeaways

• Inverse Functions Reverse Operations: Remember that inverse functions undo what the original function does.
• Steps to Find the Inverse: Replace f(x) with y, swap x and y, solve for y, and rewrite y as g(x).
• Practice Makes Perfect: The more you practice, the easier it gets to find inverse functions.

Understanding inverse functions is a fundamental skill in math, and I hope this explanation helped you grasp the concept. Keep practicing, and you'll become a pro in no time! Remember, math can be fun when you break it down step by step. Keep up the great work!