Solving Composite Function Inverse Problems For F(x) And G(x)

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Hey guys! Let's break down this interesting math problem together. We've got two functions here: f(x) which is a bit of a fraction, and g(x) which is a simple line. Our main goal is to find the inverse of the composite function (f o g), written as (f o g)⁻¹(x). Sounds a little intimidating, right? But don't worry, we'll take it step by step and make it super clear. Think of it like this: we're reverse-engineering the process of combining these functions. So grab your favorite beverage, get comfy, and let's dive in!

Understanding the Functions: f(x) and g(x)

First, let's make sure we fully understand our initial functions. We have f(x) = (x + 4) / (x - 6). The most important thing to notice here is the denominator: x - 6. We cannot let the denominator be zero (because dividing by zero is a big no-no in math!). So, x cannot be 6. This little detail is super important because it tells us about the domain of the function f(x). The domain is basically the set of all possible input values that we can plug into the function without causing any mathematical chaos. In this case, the domain of f(x) is all real numbers except for 6. This means we can put in any number we want, except 6, and the function will give us a valid output. Think of it like a machine: you can feed it anything but 6, or else the machine will jam! Next, we have g(x) = 2x - 1. This one's a lot simpler. It's just a straight line with a slope of 2 and a y-intercept of -1. For g(x), we don't have any restrictions on the input. We can plug in any real number for x, and we'll always get a valid output. So, the domain of g(x) is all real numbers. Understanding the domains of our functions is crucial because it affects the domain of the composite function we're about to create. If we try to plug in a value that's not in the domain, we'll end up with an undefined result, and that's not what we want! So, always keep an eye on those denominators and any other potential troublemakers.

Finding the Composite Function (f o g)(x)

Now that we've got a handle on f(x) and g(x) individually, let's combine them to form the composite function (f o g)(x). This notation might look a little weird, but it's actually quite straightforward. It just means we're going to plug the entire function g(x) into the function f(x). Think of it like a chain reaction: first, we apply g(x) to our input x, and then we take the result and feed it into f(x). So, wherever we see an x in f(x), we're going to replace it with the whole expression for g(x), which is 2x - 1. This is where the fun begins! Let's write it out: (f o g)(x) = f(g(x)) = f(2x - 1). Now, we'll substitute 2x - 1 into the f(x) equation: (f o g)(x) = ((2x - 1) + 4) / ((2x - 1) - 6). See how we just swapped the x in f(x) with the expression 2x - 1? Next, we need to simplify this expression to make it look cleaner and easier to work with. Let's simplify the numerator and the denominator separately. In the numerator, we have (2x - 1) + 4, which simplifies to 2x + 3. In the denominator, we have (2x - 1) - 6, which simplifies to 2x - 7. So, our composite function (f o g)(x) becomes: (f o g)(x) = (2x + 3) / (2x - 7). Awesome! We've successfully created the composite function. But we're not done yet. Remember, our ultimate goal is to find the inverse of this composite function. But before we jump into that, let's take a moment to think about the domain of this new function. Just like with f(x), we need to make sure the denominator doesn't equal zero. So, 2x - 7 cannot be zero. This means x cannot be 7/2. So, the domain of (f o g)(x) is all real numbers except for 7/2. Keeping track of these domains is crucial for understanding the behavior of our functions and avoiding mathematical pitfalls.

Finding the Inverse of the Composite Function (f o g)⁻¹(x)

Okay, the moment we've been waiting for! Now we're going to tackle the trickiest part: finding the inverse of the composite function, (f o g)⁻¹(x). Remember, finding the inverse of a function is like undoing what the function does. If a function takes an input x and gives us an output y, the inverse function takes that output y and gives us back the original input x. It's like a mathematical round trip! To find the inverse, we're going to use a little algebraic trickery. First, we'll replace (f o g)(x) with the variable y. So, we have: y = (2x + 3) / (2x - 7). Next, and this is the key step, we're going to swap the x and the y. This is the magic that starts to unravel the function and reveal its inverse. After swapping, we get: x = (2y + 3) / (2y - 7). Now, our mission is to solve this equation for y. This will give us the inverse function in terms of x. To get rid of the fraction, we'll multiply both sides of the equation by (2y - 7): x(2y - 7) = 2y + 3. Now, we'll distribute the x on the left side: 2xy - 7x = 2y + 3. Our goal is to isolate y, so let's move all the terms with y to one side and all the other terms to the other side. We'll subtract 2y from both sides and add 7x to both sides: 2xy - 2y = 7x + 3. Now, we can factor out a y from the left side: y(2x - 2) = 7x + 3. Finally, we'll divide both sides by (2x - 2) to solve for y: y = (7x + 3) / (2x - 2). Awesome! We've found the inverse function. Now, we just need to replace y with the notation for the inverse function, (f o g)⁻¹(x): (f o g)⁻¹(x) = (7x + 3) / (2x - 2). And there you have it! We've successfully found the inverse of the composite function. But, just like before, let's think about the domain of this inverse function. We need to make sure the denominator, 2x - 2, doesn't equal zero. This means x cannot be 1. So, the domain of (f o g)⁻¹(x) is all real numbers except for 1. This is a crucial piece of information that helps us fully understand the behavior of the inverse function.

Final Answer and Key Takeaways

Alright, after all that algebraic maneuvering, we've arrived at our final answer! The inverse of the composite function (f o g)(x) is: (f o g)⁻¹(x) = (7x + 3) / (2x - 2). High five! We did it! But more importantly than just getting the right answer, let's take a moment to appreciate what we've learned along the way. This problem was a fantastic workout for our algebra muscles. We tackled composite functions, inverse functions, and domain restrictions. We saw how combining functions can create new and interesting relationships, and how finding the inverse is like reversing a mathematical process. The key to solving problems like this is to break them down into smaller, more manageable steps. Don't try to do everything at once! Start by understanding the individual functions, then create the composite function, and finally, tackle the inverse. And always, always keep an eye on those domains! They're like the traffic rules of the mathematical world, guiding us and preventing us from crashing into undefined territory. So, next time you encounter a problem involving composite functions and inverses, remember the steps we took here. And remember, you've got this! You're a mathematical problem-solving rockstar!

In summary:

  • f(x) = (x + 4) / (x - 6)
  • g(x) = 2x - 1
  • (f o g)(x) = (2x + 3) / (2x - 7)
  • (f o g)⁻¹(x) = (7x + 3) / (2x - 2)