Composite Function: Find H(g(f(x))) In Quadratic Form

Hey guys! Let's dive into a super interesting problem involving composite functions. We're given three functions: $f(x)$, $g(x)$, and $h(x)$, and our mission, should we choose to accept it (and we totally do!), is to find the composite function $h(g(f(x)))$. Sounds like a mouthful, right? But don't worry, we'll break it down step by step and make it crystal clear. The ultimate goal is to express the final composite function in the quadratic form $Ax^2 + Bx + C$. So, grab your thinking caps, and let's get started!

Understanding Composite Functions

Before we jump into the nitty-gritty, let's quickly recap what composite functions are all about. Think of it like a function inside a function – a mathematical Matryoshka doll, if you will. When we write $h(g(f(x)))$, it means we're first applying the function $f$ to $x$, then taking the result and applying the function $g$ to that, and finally, we apply the function $h$ to the result of that. It's like a chain reaction, each function passing its output to the next. Understanding this sequential process is crucial for solving the problem.

So, in essence, a composite function is a function that is formed by combining two or more functions. The output of one function becomes the input of another. The notation $h(g(f(x)))$ might look intimidating, but it's just a systematic way of saying, "First do $f$, then do $g$ on the result, and finally do $h$ on that result." This concept is used extensively in calculus and other advanced math topics, so getting a solid grasp on it now is super beneficial. Remember, the order matters! $h(g(f(x)))$ is generally not the same as $f(g(h(x)))$, so we need to be meticulous in following the correct sequence.

Step 1: Find f(x)

Okay, with the basics covered, let’s tackle our specific problem. We're given $f(x) = 2x + 5$. That's our starting point, the innermost function in our composite function. There's not much to do here in terms of calculation for this step itself, but it’s important to acknowledge it. This is our foundation. We'll use this expression in the next step when we evaluate $g(f(x))$. It's like the first ingredient in our recipe – we need it to move forward. Just keeping it in mind is key, so we don’t lose track of what we're working with. Seriously, in these kinds of problems, it’s so easy to get lost in the layers, so acknowledging each step is a pro move.

Understanding the role of $f(x)$ as the initial transformation is also key to visualizing the entire process. Imagine $x$ as a raw input; $f(x)$ is the first processing step. It takes $x$, doubles it, and adds 5. The result is then fed into $g$, which will then transform it further. This step-by-step visualization helps us to understand the flow and logic behind the composition of functions. We are building a mathematical machine where each function is a component performing a specific task.

Step 2: Find g(f(x))

Now for the fun part! We need to find $g(f(x))$. Remember, $g(x) = x^2$. So, wherever we see an $x$ in the expression for $g(x)$, we're going to replace it with the entire expression for $f(x)$, which is $2x + 5$. This is the core of composite function evaluation. So, we get $g(f(x)) = (2x + 5)^2$. See how we've essentially plugged the function $f(x)$ into the function $g(x)$? Now, we need to simplify this expression. We expand the square: $(2x + 5)^2 = (2x + 5)(2x + 5)$.

Using the good ol' FOIL method (First, Outer, Inner, Last), we get: $(2x * 2x) + (2x * 5) + (5 * 2x) + (5 * 5) = 4x^2 + 10x + 10x + 25$. Combining like terms, we have $g(f(x)) = 4x^2 + 20x + 25$. Woohoo! We've successfully computed $g(f(x))$. This quadratic expression is the output of the second stage of our composite function. It's important to keep this expression in its expanded form, as it will be the input for our final function, $h(x)$. Trust me, keeping track of these intermediate results is key to avoiding errors later on.

Step 3: Find h(g(f(x)))

Alright, we're on the home stretch! Now we need to find $h(g(f(x)))$. We know that $h(x) = -2x$, and we just found that $g(f(x)) = 4x^2 + 20x + 25$. So, just like before, we're going to replace the $x$ in $h(x)$ with the entire expression for $g(f(x))$. This gives us $h(g(f(x))) = -2(4x^2 + 20x + 25)$. All that's left to do now is distribute the $-2$. This is where we multiply each term inside the parentheses by $-2$.

Distributing the -2, we get: $-2 * 4x^2 = -8x^2$, $-2 * 20x = -40x$, and $-2 * 25 = -50$. Putting it all together, we have $h(g(f(x))) = -8x^2 - 40x - 50$. Bam! We've done it! We've found the composite function $h(g(f(x)))$ and expressed it in the quadratic form $Ax^2 + Bx + C$. In this case, $A = -8$, $B = -40$, and $C = -50$. See? Not so scary after all, right?

Final Answer: h(g(f(x))) = -8x² - 40x - 50

So, there you have it! The composite function $h(g(f(x)))$ is equal to $-8x^2 - 40x - 50$. We started with three separate functions and, by carefully applying them one after another, we arrived at a single quadratic function. This whole process highlights the power and elegance of function composition. It's like building complex mathematical structures from simpler components. And the best part? You now have a solid understanding of how to tackle these types of problems. Give yourself a pat on the back – you've earned it!

To recap, we methodically worked our way from the inside out, first finding $f(x)$, then using that to find $g(f(x))$, and finally using that result to find $h(g(f(x)))$. Each step built upon the previous one, and we made sure to carefully substitute and simplify along the way. This systematic approach is the key to successfully navigating composite functions. Remember to always pay attention to the order of operations and to double-check your work to avoid those pesky little arithmetic errors. Keep practicing, and you'll become a composite function pro in no time! This is a crucial skill for higher mathematics, so investing the time to master it now will pay dividends down the road. Great job, everyone!