Function Division: A Step-by-Step Guide With Solutions

Introduction: Let's Break Down These Functions!

Hey guys! Let's dive into some math problems involving function division. We've got three functions: f(x), g(x), and h(x). Our goal is to figure out what happens when we divide these functions. Don't worry; it might sound a little scary, but it's really just about careful manipulation. Think of it like this: we're going to take one function and split it by another, piece by piece. The problems we're tackling are: f/g(x), f/h(x), and g/h(x). We'll go through each one step-by-step, using long division (also known as polynomial division) where it's needed to keep things organized and clear. Remember, the most important thing is to stay organized and pay attention to the signs – those little details can make a big difference! Before we start the problems, let's rewrite our functions in their simplest forms. This will make our calculations much easier. First, f(x) = 3x - 2x + x - 1 can be simplified to f(x) = 2x - 1. Combining the x terms, we get this result. Next, g(x) = x² + 3x² - x + 4 simplifies to g(x) = 4x² - x + 4. Again, combining like terms. Finally, h(x) = x² - 6x. Now, we're ready to roll up our sleeves and start the division. In each case, we will look at how to divide f(x) by g(x), f(x) by h(x), and finally g(x) by h(x), making it really clear what's going on, so follow along. Keep in mind that function division is a fundamental concept, and the ability to do this kind of math will give you a really good base for understanding calculus and other advanced topics later on. So, let's do it.

Problem 1: Solving for f/g(x) – Division Time!

Alright, let's tackle the first part of our mission: finding f/g(x). This means we have to divide the function f(x) = 2x - 1 by the function g(x) = 4x² - x + 4. When the degree of the denominator (the bottom part, g(x) in this case) is higher than the degree of the numerator (the top part, f(x)), our answer usually stays as a fraction. In this scenario, we don’t need long division because the denominator's highest power is greater than the numerator. Therefore, f/g(x) is simply (2x - 1) / (4x² - x + 4). We cannot simplify this any further because there are no common factors. We've done all we can for this one. The expression is what it is, and we can't break it down any further. The numerator is a linear expression (2x - 1), and the denominator is a quadratic expression (4x² - x + 4). Since there are no common factors that can be extracted from both the numerator and denominator, we end up with the result as a fraction as it is. So, that's it, that's all we got for this one, nothing more to do. Sometimes the solution to a math problem is straightforward; this is one of those cases. But, remember, the ability to recognize these kinds of simplifications will become extremely helpful as you move on to more complex problems. Keep in mind that this fraction represents the result of dividing the function f(x) by g(x) for any possible value of x for which g(x) is not zero. It's pretty neat that we can take two functions and get a brand new function by doing the math, right? Next up, we'll be dealing with our next problem in our queue!

Problem 2: Unveiling f/h(x) – More Division!

Now, let's find f/h(x), which means we're dividing f(x) = 2x - 1 by h(x) = x² - 6x. In this case, just like with the first one, the highest degree of the denominator (h(x), which is quadratic) is greater than the highest degree of the numerator (f(x), which is linear). Because of this, we again can't simplify using long division, and the answer will also remain as a fraction. So, f/h(x) equals (2x - 1) / (x² - 6x). We can’t simplify this further since there are no common factors between the numerator and the denominator. The numerator is a linear expression, and the denominator is a quadratic expression. Similar to the last problem, we do not have the luxury of simplifying it, but we still get a valid function. The most important thing in this step is to keep the terms straight, and to recognize that the original functions are now being treated in the form of a ratio, where x may take on any real value provided the denominator does not equal zero. This means that, to find a function f/h(x), we can substitute any value, and we can solve the equation to the point that we find the result. And that's it for our second problem! We took our two functions and ended up with a new function by dividing them. This is a really important process to understand because it demonstrates how new functions can come about through the process of algebraic manipulation, with this being just one example. Let's keep going! Let's see what we get when we divide g(x) by h(x).

Problem 3: Calculating g/h(x) – The Final Countdown!

Okay, team, last but not least, we're calculating g/h(x). This means dividing g(x) = 4x² - x + 4 by h(x) = x² - 6x. This one looks a bit more interesting because we've got quadratics involved. Because the numerator and denominator have the same power (quadratic), we'll need to use long division, or polynomial division, to simplify this. Let's get started. First, set up your long division problem. Write the numerator (4x² - x + 4) inside the division symbol and the denominator (x² - 6x) outside. Now, focus on dividing the first term of the numerator (4x²) by the first term of the denominator (x²). 4x² / x² = 4. Write this result on top. Multiply the result by the divisor. Multiply 4 by (x² - 6x), which gives you 4x² - 24x. Write this under your original numerator. Subtract the result of the multiplication from the numerator. Subtract (4x² - 24x) from (4x² - x + 4) to get 23x + 4. Since 23x is the same power as the divisor, this is our remainder, which means we are done. So, g/h(x) = 4 + (23x + 4) / (x² - 6x). The final answer consists of a quotient and a remainder. And that's it! We did it, guys. We successfully divided all three functions, and now you know how to approach similar problems. In this case, we went through the process of long division to get our answer. It's easy, once you get the hang of it. Remember, practicing these types of problems will help you solidify your understanding of functions and algebra. This knowledge forms a strong foundation for more advanced topics in mathematics. Keep up the great work!

Conclusion: Wrapping Things Up!

So, there you have it! We've successfully navigated the process of function division for f/g(x), f/h(x), and g/h(x). We saw that sometimes, the answer is simply a fraction because we can't simplify it further, while in other cases, we need to use long division to get to the end. Remember, the key to these problems is paying attention to the details, staying organized, and carefully applying the rules of division. With practice, you'll become more and more comfortable with these concepts. Remember that function division is a building block in mathematics, giving us the ability to work with and manipulate functions in new and exciting ways. I hope this has been a helpful guide! Keep practicing, and you'll do great! Let me know if you have any more questions – I'm here to help!