Multiplying Functions: Finding F · G and Its Domain for Let F(X) = 3x + 2 and G(X) =7x + 6

Let F(X) = 3x + 2 and G(X) = 7x + 6. Find F ·G and its domain. Multiply the functions and determine where they are defined.

Have you ever wondered what happens when two mathematical functions get together? Well, today we're going to find out! Let's take a look at F(x) = 3x + 2 and G(x) = 7x + 6. These two functions are about to collide and create something truly amazing. Are you ready to witness the birth of a new function?

First things first, let's find out what happens when we multiply F(x) and G(x). To do this, we simply need to multiply each term in F(x) by each term in G(x). This gives us:

F(x) · G(x) = (3x + 2) · (7x + 6)

Now, let's multiply these two polynomials using the distributive property:

F(x) · G(x) = 21x² + 18x + 14x + 12

Combining like terms, we get:

F(x) · G(x) = 21x² + 32x + 12

So, there you have it! The product of F(x) and G(x) is a brand new function: 21x² + 32x + 12. But, before we get too excited, we need to make sure we know the domain of this new function.

The domain of a function is the set of all possible values of x for which the function is defined. In other words, it's the set of values that we can plug into the function without getting an error. To find the domain of F(x) · G(x), we need to consider the domains of F(x) and G(x) separately.

The domain of F(x) is all real numbers, since we can plug in any value of x and get a valid output. The same is true for G(x). Therefore, the domain of F(x) · G(x) is also all real numbers.

So, there you have it! We've multiplied F(x) and G(x) to create a brand new function, and we've determined that its domain is all real numbers. Who knew math could be so exciting?

Now, let's take a closer look at what this new function actually represents. The function F(x) · G(x) is a polynomial, which means it's a function of the form ax^n + bx^(n-1) + ... + c. In this case, our polynomial is 21x² + 32x + 12.

Polynomials are incredibly useful in mathematics, since they can be used to model a wide variety of phenomena. For example, we might use a polynomial to represent the trajectory of a projectile, or the growth of a population over time. In fact, polynomials are so versatile that they show up in just about every branch of mathematics.

But, what does our polynomial actually represent? Well, without any context, it's hard to say. However, we can make a few observations based on its form.

First of all, notice that the degree of our polynomial is 2. This means that it's a quadratic function, which is a fancy way of saying that it's a parabola. If you were to graph our polynomial, you'd see that it forms a U-shape.

Secondly, notice that the coefficient of the x^2 term is positive. This means that our parabola opens upwards. In other words, as x gets larger (or smaller), the value of our function also gets larger (or smaller).

Finally, notice that our polynomial has two real roots (i.e. values of x for which the function equals zero). These roots are given by:

x = (-32 ± sqrt(32^2 - 4·21·12)) / (2·21) ≈ -1.52, -0.31

These roots correspond to the points where our parabola intersects the x-axis. In other words, they're the points where our function equals zero.

So, there you have it! We've not only multiplied F(x) and G(x) to create a brand new function, but we've also analyzed its properties and determined what it represents. Who knew math could be so fascinating?

If you're interested in learning more about polynomials and their applications, there's plenty of resources available online and at your local library. With a little bit of effort, you too can become a polynomial pro!

Introduction

Hey there, math enthusiasts! Are you ready to solve some equations and have a laugh while doing it? Well, today we're going to talk about the functions F(x) = 3x + 2 and G(x) = 7x + 6. But don't worry, I promise this won't be a snooze-fest like your high school algebra class.

What are F(x) and G(x)?

Before we dive into the juicy stuff, let's quickly review what F(x) and G(x) actually mean. In simple terms, a function is like a machine that takes in a number (x), does some calculations, and spits out a new number. So, F(x) = 3x + 2 means that if we put in a number (let's say 5), the function will multiply it by 3, add 2, and give us the answer (17). G(x) = 7x + 6 works the same way - it takes a number, multiplies it by 7, adds 6, and gives us a new number.

What is F · G?

Now, let's get to the fun part - finding F · G! This fancy notation simply means that we're going to multiply F(x) and G(x) together. Don't worry, it's not as complicated as it sounds. All we have to do is plug in G(x) wherever we see an x in F(x), like so:F · G = F(G(x)) = 3(7x + 6) + 2Now, let's simplify this equation by multiplying 3 by 7x and 3 by 6:F · G = 21x + 18 + 2F · G = 21x + 20

What is the domain of F · G?

Okay, so now we know that F · G = 21x + 20. But what does that even mean? And what's the domain? Well, the domain of a function is simply the set of all numbers that we're allowed to put into it. In other words, it's the range of numbers where the function makes sense.In this case, since we're dealing with two linear functions, the domain is all real numbers. That means we can put in any number we want and get a valid output. So, the domain of F · G is (-∞, ∞).

How do we graph F(x) and G(x)?

Now, let's take a quick break from all the math and talk about something more visual - graphs! If you're a visual learner like me, graphs are a great way to understand functions and how they relate to each other.To graph F(x) and G(x), all we have to do is plot a few points and connect them with a straight line. For example, if we plug in x = 0 into F(x), we get 2. So, one point on the graph of F(x) is (0, 2). If we plug in x = 1, we get 5. Another point is (1, 5). We can do the same thing for G(x) and get a similar graph.

What does the graph of F · G look like?

Now, let's see what happens when we multiply F(x) and G(x) together. To graph F · G, we need to plot a few points and connect them with a straight line. But since we're dealing with a new function, we don't really know what it's going to look like.Luckily, we can make an educated guess based on what we know about linear functions. Since F(x) and G(x) are both straight lines, their product should also be a straight line. In other words, the graph of F · G should be a straight line that goes through the origin (0, 0).

What are some applications of F · G?

Okay, so now we know what F · G is, what its domain is, and what its graph looks like. But why should we care? What are some real-world applications of this function?Well, one example is calculating the total cost of a phone plan. Let's say that F(x) represents the cost of data usage and G(x) represents the cost of talk time. By multiplying these two functions together, we can get a new function that tells us the total cost of the phone plan based on how much data and talk time we use.Another example is calculating the distance traveled by a car. Let's say that F(x) represents the speed of the car and G(x) represents the time elapsed. By multiplying these two functions together, we can get a new function that tells us the distance traveled by the car based on its speed and the amount of time it's been driving.

Conclusion

And there you have it, folks! We've learned about the functions F(x) and G(x), found their product (F · G), and explored some real-world applications. Don't be afraid to experiment with different functions and see what kind of crazy graphs you can come up with. And remember, math doesn't have to be boring - let's inject some humor and personality into it!

Buckle up for the math ride of your life!

Do you dread math problems? Well, fear no more! Let’s F(X) and G(X) our way to greatness with this fun-i-cal problem. Warning: may cause extreme brain stimulation. But who says numbers can’t be enjoyable? They just need a little humor injection!

The Dynamic Duo

Meet F(X) and G(X), the dynamic duo you never knew you needed. Until now! F(X) = 3x + 2 and G(X) = 7x + 6. Don’t let the letters and symbols intimidate you. Think you’re bad at math? Think again. Maybe you’re just too serious.

The Ultimate Question

Now, for the ultimate question. What is F ·G and its domain? Here’s a math problem that won’t make you break a sweat (too much).

First, let’s find F ·G. It’s as easy as combining the two equations and multiplying:

F ·G = (3x + 2)(7x + 6)

Expand the brackets:

F ·G = 21x² + 36x + 14x + 12

Simplify:

F ·G = 21x² + 50x + 12

Voila! F ·G is 21x² + 50x + 12.

Now, for the domain. Don’t worry, it’s not as complicated as it sounds. The domain is simply the set of all values that x can take. In this case, x can be any real number because there are no restrictions on the variables in either equation.

The Grand Finale

And there you have it, folks! F ·G = 21x² + 50x + 12 with a domain of all real numbers. Why settle for simple arithmetic when you can have a full-on math adventure? So next time you’re faced with a math problem, remember, F(X) and G(X) are about to become your new best friends!

The Hilarious Tale of F(X) and G(X)

Once upon a time, there were two mathematical functions:

Let me introduce you to F(X) = 3x + 2 and G(X) = 7x + 6. F(X) was a sassy little function who always knew how to add a little spice to any equation. G(X), on the other hand, was a bit more serious and always followed the rules.

Finding the Product of F(X) and G(X)

One day, while F(X) and G(X) were hanging out in their domain, they decided to team up and find their product. They called it F · G, which is just a fancy way of saying that they were going to multiply themselves together.

To find F · G, we simply need to multiply the two functions: (3x + 2) · (7x + 6).
This gives us: 21x² + 24x + 12.

Wow, F(X) and G(X) make quite the dynamic duo!

The Domain of F · G

Now that they've found their product, F(X) and G(X) wanted to figure out their domain. The domain is just a fancy way of saying all the values of x that make sense for the function.

Since both F(X) and G(X) are defined for all real numbers, their product F · G is also defined for all real numbers. In other words, the domain of F · G is (-∞, ∞).
But wait, there's more! Since F(X) and G(X) are both linear functions (meaning they make a straight line when graphed), their product F · G is also a quadratic function (meaning it makes a parabola when graphed).
This means that the range of F · G might be limited, depending on the coefficients of the quadratic.

Who knew math could be so entertaining? F(X) and G(X) certainly know how to add some humor to the world of functions!

The End

Closing Time: Goodbye and Good Luck Finding Your Perfect Math Match!

Well, folks, we've reached the end of our journey together. We've explored the world of mathematical functions and learned how to find F·G and its domain using the power of algebra. It's been a wild ride, but now it's time to say goodbye.

Before we part ways, let's take one final look at our two favorite functions: Let F(X) = 3x + 2 And G(X) =7x + 6. These two have been through a lot together, but they always manage to come out on top.

So, without further ado, let's find F·G and its domain. We know that F(X) = 3x + 2 and G(X) = 7x + 6, so F·G = (3x + 2) * (7x + 6).

Now, we could do the math and simplify this equation, but let's face it, we're all a little tired of doing equations. Instead, let's focus on the domain.

The domain of any function is the set of values that the function can take. In other words, it's the set of all possible inputs that will give us a valid output.

In the case of F·G, the domain is simply the set of all real numbers. Why? Because there are no restrictions on what values x can take. You can put in any number you like, and you'll get a valid result.

So, there you have it, folks. F·G and its domain. But before I go, I want to leave you with one final thought:

Mathematical functions are a lot like dating. You have to find the right match for you. Sometimes it's love at first sight, and other times it takes a little work. But when you find that perfect match, everything just clicks.

So, keep searching for your perfect math match. Keep exploring the world of functions and equations. And most importantly, never give up on finding the solution.

And with that, I bid you adieu. It's been a pleasure sharing my love of math with you all. Good luck on your mathematical journey, and may all your functions be smooth and elegant.

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