Finding F(x): A Step-by-Step Guide For Beginners

Hey guys! Ever stumbled upon a math problem that looks a bit intimidating at first glance? Well, don't worry, because today we're going to break down a common type of problem: finding a function when you're given its composition with another function. Specifically, we'll tackle the problem of finding f(x) when we know g(x) = x + 1 and (fog)(x) = x² + 3x + 1. Sounds a bit like a secret code, right? But trust me, with a few simple steps, we can crack it! This kind of problem pops up quite a bit in algebra, so understanding it will definitely give you a leg up. The key here is understanding function composition. Think of it like a machine: you put something in, and it spits something else out. In this case, we're dealing with two machines, f and g, working together. The notation (fog)(x) means we're first applying the g function to x, and then applying the f function to the result. It's super important to remember the order!

Let's get started with a clear understanding. We have two main pieces of information: the expression for g(x) and the expression for the composite function (fog)(x). Our goal is to figure out what the function f(x) actually is. It's like reverse engineering a recipe: we know the final dish and one of the ingredients, and we need to figure out the other ingredients. First of all, let's define function composition more clearly. When we write (fog)(x), it is interpreted as f(g(x)). This means that we substitute the expression for g(x) into f(x). To solve this problem, we use the concept of substitution. We're going to use the known value of g(x) (which is x + 1) and substitute it into the composite function (fog)(x). This will help us unravel the mystery of what f(x) does.

Step-by-Step Solution to Find f(x)

Alright, let's get our hands dirty and solve this thing, shall we? Here's the breakdown, step by step, to get us to the solution. It's like following a treasure map, and the answer is the treasure! Firstly, let's start by rewriting the composition (fog)(x) using the definition. We know that (fog)(x) is the same as f(g(x)). Because g(x) = x + 1, we can rewrite the composite function as f(x + 1) = x² + 3x + 1. This is super important because now we can see how f is acting on x + 1. Notice that the x inside f isn't just x anymore; it's x + 1. Our task is to find out what f does to whatever is inside its parentheses. Now comes the clever part: substitution. The next step is to make a substitution to simplify things. Let's say u = x + 1. If u = x + 1, then x = u - 1. This is a simple algebraic manipulation, and it allows us to rewrite everything in terms of u. So, everywhere we see x + 1, we'll replace it with u, and everywhere we see x, we'll replace it with u - 1. It's like giving the equation a makeover!

Let's sub the value in. Substituting u into the f(x + 1) = x² + 3x + 1 equation, we'll get f(u) = (u - 1)² + 3(u - 1) + 1. Now we simplify it to make it easier to read. Now it is just a matter of expanding and simplifying the right side of the equation. We can do this by expanding f(u) = (u - 1)² + 3(u - 1) + 1. Expanding (u - 1)², we get u² - 2u + 1. Next, we expand 3(u - 1) to get 3u - 3. Putting it all together, we now have f(u) = u² - 2u + 1 + 3u - 3 + 1. Then, we simplify the expression, so we combine like terms (the terms with u and the constant terms). We combine -2u and 3u to get u. And then, we combine the constants 1, -3, and 1 to get -1. This simplifies our equation to f(u) = u² + u - 1. And finally, we replace u with x. Since the variable name doesn't really matter, we can replace u with x to write the function in the standard form of f(x). So, our final answer is f(x) = x² + x - 1. Bam! We did it, guys! We found the function f(x).

Checking the Solution

Now that we've found our solution, it's always a good idea to check if it's correct. Verification is the key, so we're going to make sure our answer works. Let's plug our answer back into the original composition to see if it matches the one we were given. We'll calculate (fog)(x) using the function f(x) = x² + x - 1 and g(x) = x + 1.

First, start by finding f(g(x)). Knowing that g(x) = x + 1, we want to find f(x + 1). Substituting (x + 1) into the function f(x) = x² + x - 1, we get f(x + 1) = (x + 1)² + (x + 1) - 1. Then, we expand (x + 1)², giving us x² + 2x + 1. Then, we expand and simplify the entire equation. So, f(x + 1) = x² + 2x + 1 + x + 1 - 1. Combining like terms, we get f(x + 1) = x² + 3x + 1.

And guess what, guys? This is exactly the same as the given (fog)(x)! That confirms that our function f(x) = x² + x - 1 is correct! This process of checking your answer is a super important habit to develop in math. It can help you catch any mistakes you might have made along the way and boost your confidence in your final answer. Plus, it gives you a deeper understanding of how the functions interact with each other. Always remember to verify your answers whenever possible.

Conclusion

So, there you have it, folks! We've successfully found the function f(x) when given g(x) and (fog)(x). This process showcases the power of function composition and how you can use it to solve complex problems. It's like a puzzle, and we've just put the pieces together. This whole process might seem daunting at first, but with practice, you'll get the hang of it. Break it down into smaller steps, and you can tackle any math problem. Keep in mind the key concepts like substitution, and always remember to double-check your work. So, the next time you encounter a function composition problem, you'll know exactly what to do! And that, my friends, is the beauty of mathematics – breaking down complex problems into manageable steps and finding elegant solutions. Keep practicing, keep learning, and keep exploring the fascinating world of math!