Finding H(x) When H(x) = F(x) - G(x) With Given Quadratic Functions
Hey guys! Today, we're diving into a super fun math problem that involves finding a new function, h(x), by subtracting two other functions, f(x) and g(x). It might sound a little complicated at first, but trust me, it's actually pretty straightforward once you get the hang of it. We're given two quadratic functions, which are just fancy names for functions with an term, and our mission is to subtract one from the other. So, let’s jump right in and break it down step by step. This is a fundamental concept in algebra, and understanding it will really help you tackle more advanced problems later on. So, buckle up, and let's get started!
Understanding the Problem
Okay, first things first, let’s make sure we all understand what the problem is asking. We have two functions:
- f(x) = 8x² - 2x + 3
- g(x) = 12x² + 4x - 3
Our goal is to find a new function, h(x), which is defined as the difference between f(x) and g(x). In math speak, that’s:
h(x) = f(x) - g(x)
Basically, what we need to do is subtract the entire function g(x) from the entire function f(x). Sounds simple enough, right? But, it's super important to pay close attention to the signs and make sure we're combining the like terms correctly. Think of it like this: we're taking all the ingredients of f(x) and subtracting all the ingredients of g(x). This means each term in g(x) needs to have its sign flipped when we subtract it. We'll go through this in detail in the next section, so don't worry if it seems a bit abstract right now. The key takeaway here is understanding that we're not just subtracting numbers, we're subtracting entire functions, which means we need to handle each term individually. Understanding this concept is crucial for mastering function operations, so let's move on and see how to actually perform the subtraction!
Step-by-Step Solution
Alright, let's get down to business and solve this thing! We're going to take it slow and steady, making sure we don’t miss any steps. This is where the magic happens, guys!
Step 1: Write Down the Functions
First, let's write down our functions so we can see them clearly:
f(x) = 8x² - 2x + 3
g(x) = 12x² + 4x - 3
This might seem like a no-brainer, but trust me, writing things down clearly is half the battle in math. It helps us keep track of everything and reduces the chances of making silly mistakes. Think of it as laying out all your ingredients before you start cooking – you want to make sure you have everything you need and that it's all in the right place. In this case, our "ingredients" are the terms of the functions, and we want to make sure we have them written down correctly before we start subtracting.
Step 2: Set Up the Subtraction
Now, let's set up the subtraction. We know that h(x) = f(x) - g(x), so we'll write that out with the actual functions:
h(x) = (8x² - 2x + 3) - (12x² + 4x - 3)
Notice the parentheses? Those are super important! They tell us that we're subtracting the entire function g(x), not just the first term. It’s like saying, "Hey, we're taking away this whole group of stuff, not just one piece of it." The parentheses remind us that we need to distribute the negative sign to every term inside the second set of parentheses. This is a common mistake people make, so always double-check that you've got those parentheses in place. Think of it like a protective shield around g(x), making sure we subtract every part of it correctly.
Step 3: Distribute the Negative Sign
This is where things get a little tricky, but don't worry, we got this! We need to distribute the negative sign (the minus sign) in front of the parentheses to each term inside the parentheses of g(x). This means we're going to change the sign of each term in g(x). It’s like we’re flipping the switch on each term, turning positives into negatives and negatives into positives. So, let's do it:
h(x) = 8x² - 2x + 3 - 12x² - 4x + 3
See what happened? The +12x² became -12x², the +4x became -4x, and the -3 became +3. It's like we're taking the opposite of each term in g(x). This is a crucial step because if we forget to distribute the negative sign, we'll end up with the wrong answer. Think of it like carefully unwrapping a present – you need to make sure you take off all the layers to get to the good stuff inside. In this case, the "layers" are the parentheses, and we need to "unwrap" them by distributing the negative sign.
Step 4: Combine Like Terms
Okay, we're in the home stretch now! The next step is to combine like terms. Like terms are terms that have the same variable and exponent. For example, 8x² and -12x² are like terms because they both have x². Similarly, -2x and -4x are like terms because they both have x. And 3 and 3 are like terms because they're both constants (just numbers). Combining like terms is like sorting your socks – you group together all the ones that match. So, let's group our like terms together:
h(x) = (8x² - 12x²) + (-2x - 4x) + (3 + 3)
We've just rearranged the terms so that the like terms are next to each other. This makes it easier to see what we need to combine. Now, let's actually combine them by adding or subtracting their coefficients (the numbers in front of the variables):
- 8x² - 12x² = -4x²
- -2x - 4x = -6x
- 3 + 3 = 6
It's like we're adding up the same kind of things. We have 8 of something and we take away 12, so we're left with -4 of that thing. We owe 2 of something and we owe another 4, so we owe a total of 6. And we have 3 of something and we add another 3, so we have 6. Make sense?
Step 5: Write the Final Answer
Finally, we can put it all together to get our final answer for h(x):
h(x) = -4x² - 6x + 6
And there you have it! We've successfully found h(x) by subtracting g(x) from f(x). High five! This is our new function, and it represents the difference between the two original functions. Remember, math is like building with blocks – we started with the individual functions, then we combined them in a specific way (subtraction) to create something new. In this case, our new creation is h(x), and it's all thanks to our careful steps and attention to detail.
Conclusion
So, guys, we did it! We successfully found h(x) = -4x² - 6x + 6 by subtracting the function g(x) from f(x). Remember the key steps: write down the functions, set up the subtraction with parentheses, distribute the negative sign, combine like terms, and then write out the final answer. Each step is like a piece of the puzzle, and when you put them all together, you get the solution!
This whole process might seem like a lot at first, but the more you practice, the easier it will become. Think of it like learning to ride a bike – you might wobble a bit at the beginning, but eventually, you'll be cruising along smoothly. The important thing is to keep practicing and don't be afraid to make mistakes. Mistakes are just learning opportunities in disguise!
Understanding how to subtract functions is a fundamental skill in algebra, and it's going to be super helpful as you move on to more advanced topics. You'll encounter function operations like this in calculus, pre-calculus, and even in real-world applications like physics and engineering. So, pat yourself on the back for tackling this problem, and keep up the awesome work! You're one step closer to becoming a math whiz. And remember, if you ever get stuck, just break the problem down into smaller steps and take it one piece at a time. You got this! Keep practicing, keep exploring, and keep having fun with math! You're doing great!
