Calculate Fg(1) For F(x) = 8/x^2 And G(x) = 4x^3

Hey math whizzes! Today, we're diving into the awesome world of functions and tackling a super common problem: finding the product of two functions, specifically $fg(1)$. We're given two cool functions, $f(x) = \frac{8}{x^2}$ and $g(x) = 4x^3$. Our mission, should we choose to accept it, is to figure out what the value of their product is when $x$ is equal to 1. This might sound a bit tricky, but trust me, guys, it's all about breaking it down step-by-step. We'll go through the process, explain each part, and by the end, you'll be a $fg(1)$ pro! So, grab your calculators, maybe a snack, and let's get this math party started! It's all about understanding how these functions interact and what their combined output looks like at a specific point. We'll explore the concept of function multiplication and then apply it to our specific case. Get ready to flex those brain muscles!

Understanding Function Multiplication: The Basics

Alright, let's chat about what it means to multiply two functions, denoted as $fg(x)$. When we see $fg(x)$, it's a shorthand way of saying "the function $f$ multiplied by the function $g$." So, to find the expression for $fg(x)$, all we do is multiply the expressions for $f(x)$ and $g(x)$ together. In mathematical terms, this looks like: $fg(x) = f(x) \times g(x)$ It's pretty straightforward, right? We just take whatever $f(x)$ is equal to and multiply it by whatever $g(x)$ is equal to. Think of it like this: if $f(x)$ is a recipe for cookies and $g(x)$ is a recipe for icing, then $fg(x)$ is the recipe for the whole decorated cookie! We're combining their essence. Now, for our specific problem, we have $f(x) = \frac{8}{x^2}$ and $g(x) = 4x^3$. So, to find the expression for $fg(x)$, we'll substitute these into our formula: $fg(x) = \left(\frac{8}{x^2}\right) \times (4x^3)$ See? We're just plugging in the given expressions. The next step, which we'll get to shortly, is to simplify this product. Simplifying is key in math, it helps us see the core relationship between the functions more clearly. We'll be using some basic exponent rules here, so if those are a little rusty, don't sweat it; we'll recap them as we go. The goal is to get a single, neat expression for $fg(x)$ before we move on to evaluating it at a specific point. This multiplication step is fundamental to understanding composite functions and other advanced function operations, so mastering it is a big win for your math journey!

Calculating the Product Expression $fg(x)$

Now that we know how to multiply functions, let's actually do it for our specific problem. We have $f(x) = \frac{8}{x^2}$ and $g(x) = 4x^3$. We want to find $fg(x)$, which means we multiply $f(x)$ by $g(x)$: $fg(x) = f(x) \times g(x) = \left(\frac8}{x^2}\right) \times (4x^3)$ To simplify this, we can rewrite it as $fg(x) = \frac{8 \times 4x^3x^2}$ Now, we multiply the constants in the numerator $8 \times 4 = 32$. So, we have: $fg(x) = \frac{32x^3x^2}$ Here's where those exponent rules come into play, guys! Remember that when you divide terms with the same base, you subtract their exponents. So, $x^3$ divided by $x^2$ is $x^{3-2} = x^1$, which is just $x$. Applying this to our expression $fg(x) = 32x^{(3-2) = 32x^1 = 32x$ And there you have it! The simplified expression for the product of our functions $f(x)$ and $g(x)$ is $fg(x) = 32x$. Isn't that neat? We started with two separate expressions, and after multiplying and simplifying, we ended up with a very simple linear function. This simplification process is super important because it makes the next step, evaluating the function, a breeze. If we had tried to plug $x=1$ into the unsimplified expression, it would have been way more work and prone to errors. So, always simplify first when you can. This really shows the power of algebraic manipulation and understanding exponent rules. It's like finding a shortcut on a long road trip – makes the journey much smoother and faster!

Evaluating $fg(1)$: The Final Step

Okay, we've done the heavy lifting! We found that the product function is $fg(x) = 32x$. Now, the question asks us to find $fg(1)$. This simply means we need to substitute $x=1$ into our simplified expression for $fg(x)$. So, wherever we see an $x$ in $32x$, we're going to replace it with the number 1: $fg(1) = 32 \times (1)$ And, as we all know, 32 multiplied by 1 is just 32! $fg(1) = 32$ Boom! We've found our answer. So, for the functions $f(x) = \frac{8}{x^2}$ and $g(x) = 4x^3$, the value of $fg(1)$ is 32. It's as simple as that! This whole process illustrates a fundamental concept in algebra: breaking down complex problems into smaller, manageable steps. First, we understood what function multiplication meant. Second, we calculated the product expression by multiplying the given functions and simplifying using exponent rules. Finally, we evaluated the resulting function at the specified value of $x$. Each step built upon the previous one, leading us smoothly to the final answer. This methodical approach is crucial not just in math but in problem-solving in general. Always remember to simplify before you evaluate if possible – it saves time and reduces errors. You guys totally crushed this! Keep practicing, and you'll become even more comfortable with these types of problems. The satisfaction of solving it yourself is awesome, right?

Alternative Method: Evaluate First, Then Multiply

Now, while simplifying first is usually the best approach, let's explore an alternative way to solve this just to show that math often has multiple paths to the same destination, guys! This method involves evaluating $f(x)$ and $g(x)$ individually at $x=1$ before multiplying them. So, let's start by finding $f(1)$: $f(1) = \frac8}{(1)^2}$ Since $1^2 = 1$, this becomes $f(1) = \frac{8{1} = 8$ Easy peasy! Now, let's find $g(1)$: $g(1) = 4(1)^3$ Since $1^3 = 1$, this becomes: $g(1) = 4 \times 1 = 4$ We have $f(1) = 8$ and $g(1) = 4$. Remember that $fg(1)$ means $f(1) \times g(1)$. So, we just multiply our individual results: $fg(1) = f(1) \times g(1) = 8 \times 4$ And $8 \times 4$ equals... 32! See? We got the exact same answer, 32. This method can sometimes be quicker if you only need to find the value of the product at a single point, and the functions are simple enough to evaluate without much hassle. However, if you needed to find $fg(x)$ for several different values of $x$, deriving the general expression $fg(x) = 32x$ first would be much more efficient. It’s like choosing between walking directly to a single destination or charting a whole map of the city – the latter is better if you plan to visit multiple places. Both methods are valid, and understanding them gives you more tools in your mathematical toolbox. It's always good to know different ways to approach a problem; it enhances your understanding and flexibility!

Conclusion: Mastering Function Products

So there you have it, math enthusiasts! We've successfully tackled the problem of finding $fg(1)$ for the functions $f(x)=\frac{8}{x^2}$ and $g(x)=4 x^3$. We explored two effective methods: first, by finding the product function $fg(x) = 32x$ and then evaluating it at $x=1$ to get 32, and second, by evaluating $f(1)$ and $g(1)$ separately and then multiplying their results, which also yielded 32. Both paths led us to the same awesome conclusion: $fg(1) = 32$. This exercise highlights the fundamental principles of function multiplication and evaluation. It's crucial to understand that $fg(x)$ is simply the product of the individual function expressions, and $fg(a)$ is the value of that product at a specific point $x=a$. Remember the power of simplification – it often makes subsequent calculations much easier and less error-prone. And hey, don't forget that there can be multiple ways to solve a problem; being flexible in your approach is a sign of a strong mathematical mind. Keep practicing these concepts, guys, because the more you do, the more natural it will feel. Whether you're dealing with basic functions or more complex ones, the strategies we used today – breaking down the problem, applying relevant rules (like exponent rules!), and checking your work – will serve you well. You've got this! Keep exploring, keep learning, and keep having fun with math! The journey of understanding functions is exciting, and you're well on your way to mastering it.