Find The Product Of Functions F(x) And G(x)

Hey guys! Today, we're diving into a fun math problem where we need to find the product of two functions. Specifically, we're given the functions $f(x) = 2x + 3$ and $g(x) = -2x$, and our mission is to figure out what f(x) ullet g(x) equals. Don't worry, it's not as scary as it sounds! We'll break it down step by step so you can totally nail it. Let's jump right in and get those math gears turning!

Understanding the Problem

Before we get our hands dirty with the actual calculation, let's make sure we're all on the same page about what the problem is asking. We have two functions, $f(x)$ and $g(x)$, and we need to find their product. In math lingo, that means we need to multiply the two functions together. So, when you see f(x) ullet g(x), just think of it as $f(x)$ times $g(x)$. Remember, functions are like little machines: you put in a value (in this case, x), and they spit out another value based on the rule they follow. Here, f(x) follows the rule “multiply x by 2 and then add 3,” while g(x) follows the rule “multiply x by -2.” To find the product, we're going to take the expressions for these rules and multiply them. Think of it like combining the actions of these two machines. We're not plugging in a specific number for x just yet; we're finding a general expression that represents the result of this multiplication, which means our answer will still have x in it. Got it? Great! Now, let’s roll up our sleeves and get multiplying.

Step-by-Step Solution

Okay, let's get down to business and find the product of our functions. Here’s how we’re going to do it, step by step, so you can follow along easily. First, we need to write out what we're trying to find: f(x) ullet g(x). Since we know what $f(x)$ and $g(x)$ are, we can substitute their expressions into this product. So, we have (2x + 3) ullet (-2x). Now, this looks like a job for the distributive property! Remember, the distributive property tells us that we need to multiply each term inside the first set of parentheses by each term in the second set. In simpler terms, we need to multiply $-2x$ by both $2x$ and $+3$. Let's start with multiplying $-2x$ by $2x$. That gives us $-4x^2$. Why? Because $2$ times $-2$ is $-4$, and $x$ times $x$ is $x^2$. Next, we need to multiply $-2x$ by $+3$. That gives us $-6x$. Again, simple multiplication: $-2$ times $3$ is $-6$, and we just tag the $x$ along. Now, we combine these two results. We have $-4x^2$ and $-6x$. Putting them together, we get $-4x^2 - 6x$. And that's it! We've found the product of $f(x)$ and $g(x)$.

Multiplying the Functions

Alright, let's walk through the multiplication process step-by-step to make sure we've got it crystal clear. We're starting with f(x) ullet g(x) = (2x + 3)(-2x). The key here is to remember the distributive property. It's like making sure everyone at the party gets a piece of the pizza. In this case, $-2x$ is the pizza, and it needs to be "distributed" to both $2x$ and $+3$ inside the parentheses. First, we take $-2x$ and multiply it by $2x$. This is where your knowledge of exponents comes in handy. Remember, when you multiply variables with exponents, you add the exponents. So, $-2x$ times $2x$ is the same as $(-2 imes 2) imes (x imes x)$. That's $-4$ times $x^2$, or $-4x^2$. Easy peasy, right? Next up, we multiply $-2x$ by $+3$. This is a bit simpler. $-2$ times $3$ is $-6$, and we just bring the $x$ along for the ride. So, $-2x$ times $+3$ is $-6x$. Now we have two terms: $-4x^2$ and $-6x$. The final step is to combine these terms. Since they are not like terms (one has $x^2$ and the other has $x$), we can't simplify them further. We just write them together: $-4x^2 - 6x$. This is our final product! You see, multiplying functions is just like any other algebraic multiplication, as long as you remember the distributive property and the rules for multiplying variables with exponents. Keep practicing, and you'll become a pro in no time!

Identifying the Correct Option

Now that we've crunched the numbers and found that f(x) ullet g(x) = -4x^2 - 6x, the next step is to match our result with the given options. This is where you put on your detective hat and carefully compare what we've calculated with the answer choices. We're looking for the option that exactly matches $-4x^2 - 6x$. Remember, it's crucial to pay attention to the signs (positive or negative) and the coefficients (the numbers in front of the x terms). One wrong sign or number, and you'll end up with the wrong answer! So, let's take a look at the options. Option A says $4x^2 + 6x$. Nope, the signs are all wrong. Option B says $-4x^2 + 6x$. Close, but no cigar – the sign of the $6x$ term is incorrect. Option C says $4x^2 - 6x$. Again, incorrect signs. And finally, Option D says $-4x^2 - 6x$. Bingo! This is an exact match for what we calculated. So, the correct answer is Option D. See how important it is to work carefully and double-check your work? A small mistake in the calculation can lead you to a completely different answer choice. But with a clear understanding of the steps and a keen eye for detail, you can nail these types of problems every time.

Common Mistakes to Avoid

Alright, let’s talk about some common pitfalls that students often stumble into when tackling problems like this. Being aware of these mistakes can help you steer clear of them and boost your confidence in getting the right answer. One of the biggest culprits is messing up the signs. Remember, when you're multiplying, a negative times a positive is a negative, and a negative times a negative is a positive. It's super easy to drop a negative sign or get the signs mixed up, especially when you're working quickly. So, always double-check your signs! Another common error is forgetting to distribute properly. We talked about the distributive property being like making sure everyone gets a piece of the pizza. If you only multiply $-2x$ by $2x$ but forget to multiply it by $+3$, you're leaving someone out, and your answer will be incomplete. Make sure you multiply each term inside the parentheses by the term outside. Exponent errors can also trip you up. When you multiply $x$ by $x$, you're adding the exponents (which are both 1 in this case), so you get $x^2$. But sometimes, people mistakenly multiply the exponents instead. Remember the rule: when multiplying powers with the same base, you add the exponents. Finally, a simple arithmetic mistake can throw everything off. Whether it's miscalculating $2 imes -2$ or making a mistake in adding or subtracting terms, even small errors can lead to the wrong answer. The best way to avoid these mistakes? Practice, practice, practice! The more you work through these types of problems, the more natural the steps will become, and the less likely you are to make these common errors.

Practice Makes Perfect

Okay, you've seen the solution, we've broken it down, and we've talked about common mistakes. Now it's your turn to shine! The best way to really master this type of problem is to practice, practice, practice. Math isn't a spectator sport, guys. You can't just watch someone else do it and expect to become an expert. You need to get your hands dirty and work through the problems yourself. So, grab a pencil and some paper, and let's try a similar example. How about this: if $f(x) = 3x - 2$ and $g(x) = -x$, what is f(x) ullet g(x)? Work through the same steps we used in the previous problem: substitute the expressions for $f(x)$ and $g(x)$, apply the distributive property, and combine like terms. Don't be afraid to make mistakes – that's how we learn! If you get stuck, go back and review the steps we outlined earlier. And if you get the answer right, awesome! You're on your way to becoming a function-multiplying whiz. The more you practice, the faster and more confidently you'll be able to solve these problems. You might even start to enjoy them (okay, maybe that's a stretch for some of you, but who knows!). So, keep at it, and remember, every problem you solve is a step closer to mastering algebra. You've got this!

Conclusion

Alright guys, we've reached the end of our function-multiplying adventure! We started with the problem of finding f(x) ullet g(x) given $f(x) = 2x + 3$ and $g(x) = -2x$, and we walked through the entire solution step by step. We learned that multiplying functions is all about using the distributive property and carefully combining like terms. We also talked about common mistakes to watch out for, like sign errors and forgetting to distribute properly. And most importantly, we emphasized the power of practice in mastering these skills. Remember, math is like a muscle – the more you exercise it, the stronger it gets. So, don't be discouraged if you find these problems challenging at first. Keep practicing, and you'll see your skills improve over time. You now know how to find the product of functions, identify the correct answer choice, and avoid common pitfalls. Go forth and conquer those algebra problems! And remember, if you ever get stuck, don't hesitate to ask for help. There are tons of resources out there, from textbooks and online tutorials to teachers and classmates who are happy to lend a hand. Keep up the great work, and I'll catch you in the next math adventure!