Finding H(1) Given Composite Functions F(x) And G(x)

Hey guys! Ever stumbled upon a math problem that looks like a puzzle? Well, today we're diving into one that involves composite functions. Don't worry, it's not as scary as it sounds! We're going to break it down step-by-step, so you'll be a pro in no time. Our mission today is to find the value of h(1), given that f(x) = x², g(x) = 8x + 4, and h(x) is the composite function (g ∘ f)(x). Let's get started!

Understanding Composite Functions

Before we jump into solving the problem, let's quickly recap what composite functions are all about. Think of it like this: you have two machines, f and g. The output of machine f becomes the input of machine g. That's essentially what a composite function is! In mathematical terms, (g ∘ f)(x) means we first apply the function f to x, and then we apply the function g to the result. It's written as g(f(x)). This notation tells us the order of operations: f first, then g. Understanding this order is crucial for solving these types of problems. We need to evaluate the inner function f(x) before we can plug that result into the outer function g(x). Many common errors in composite function problems arise from performing these operations in the wrong order, so always double-check your steps to ensure you're applying the functions in the correct sequence. With a solid grasp of this concept, we're well-equipped to tackle the problem at hand and find the value of h(1) efficiently.

Step-by-Step Solution

Okay, let's get our hands dirty and solve this problem! We know that h(x) = (g ∘ f)(x), which means h(x) = g(f(x)). Our goal is to find h(1), so we need to figure out what happens when we plug in 1 for x.

1. Find f(1)

First, we need to find the value of f(1). We're given that f(x) = x². So, to find f(1), we simply substitute x with 1:

f(1) = (1)² = 1

Easy peasy, right? Now we know that f(1) = 1. This result is key because it becomes the input for our next step. Remember, in composite functions, we work from the inside out. We've successfully evaluated the inner function f(x) at x = 1, and now we're ready to use this output as the input for the outer function, g(x). This step highlights the importance of understanding the order of operations in composite functions; evaluating f(1) first is crucial for correctly determining the final value of h(1). Without this foundational step, we wouldn't be able to proceed accurately. So, let's carry this momentum forward and see what happens when we plug this value into g(x).

2. Find g(f(1))

Now that we know f(1) = 1, we can find g(f(1)), which is the same as g(1). We're given that g(x) = 8x + 4. So, to find g(1), we substitute x with 1:

g(1) = 8(1) + 4 = 8 + 4 = 12

Fantastic! We've found that g(1) = 12. This means that g(f(1)) = 12. Remember, this is the crucial step where we combine the results of our previous calculations. We first found f(1), and then we used that result as the input for g(x). This process perfectly illustrates the essence of composite functions – the output of one function becomes the input of another. The calculation itself is straightforward, but the underlying concept is what's important here. We're not just plugging in numbers; we're following a sequence of operations dictated by the composition of the functions. This careful, step-by-step approach ensures that we arrive at the correct answer. Now that we've successfully navigated this step, we're just one step away from our final solution!

3. Determine h(1)

Since h(x) = g(f(x)), we've already found that g(f(1)) = 12. Therefore:

h(1) = 12

And there you have it! The value of h(1) is 12. We did it! This final answer represents the culmination of all our previous work. We started by understanding the definition of composite functions, then systematically evaluated f(1) and g(f(1)), and finally arrived at the solution for h(1). This process highlights the importance of a structured approach to problem-solving. By breaking down the complex problem into smaller, manageable steps, we were able to tackle it effectively and confidently. Remember, in mathematics (and in life!), taking things one step at a time can make even the most daunting challenges seem achievable. So, let's celebrate this small victory and use this newfound knowledge to conquer even more mathematical puzzles!

Conclusion

So, there you have it, guys! We've successfully found the value of h(1) by understanding composite functions and breaking down the problem into smaller steps. Remember, the key is to work from the inside out, evaluating f(x) first and then using that result as the input for g(x). This concept of composite functions pops up everywhere in math, so mastering it is a huge win. We navigated through the problem step-by-step, highlighting the importance of understanding the order of operations and the underlying concept of composite functions. By systematically evaluating f(1) and then using that result to calculate g(f(1)), we arrived at the final solution for h(1). This structured approach not only helped us solve the problem accurately but also reinforced the importance of clear and logical thinking in mathematics. Keep practicing, and you'll be solving these problems like a pro in no time! Now that you've got this down, you're ready to tackle even more challenging problems. Keep practicing, keep exploring, and most importantly, keep having fun with math! You've got this!