Solving For X: When Function Subtraction Equals Zero

by Andrew McMorgan 53 views

Hey guys! Let's dive into a cool math problem. We're gonna figure out the value of x when the subtraction of two functions equals zero. This is a common type of question, so understanding it will definitely boost your math skills. We'll break it down step-by-step, making it super easy to follow. Get ready to flex those math muscles!

Understanding the Problem: The Basics of Function Subtraction

Alright, first things first. What exactly does the question ask? We're given two functions, f(x) = 16x - 30 and g(x) = 14x - 6. The expression (f - g)(x) means we need to subtract the function g(x) from the function f(x). Then, we have to find the value of x that makes the result of this subtraction equal to zero. Think of it like a puzzle – we need to find the missing piece, which in this case, is the value of x. This concept is fundamental in algebra, and it's super important to understand. Let's start with function subtraction. This is where we take one function and subtract another. When we write (f - g)(x), it means f(x) - g(x). We're basically combining two functions into one by subtracting their expressions. The goal is to simplify, and then solve for x. Remember that the order matters here. We're taking f and subtracting g from it. Getting this straight from the start will avoid any future misunderstandings. So, if we mess up the order, the answer will be totally wrong. Now, we are ready to find the correct value.

Let’s start to break down this problem. First, we need to understand the function subtraction part. When we have (f - g)(x), it represents subtracting g(x) from f(x). So, the first step is really about setting up the problem correctly. Next, we need to carefully apply the subtraction. We’re going to subtract the expression of g(x) from the expression of f(x). This step is crucial. This is where you might want to break out a pencil and paper to keep things clean and organized. Remember to pay close attention to the signs – especially when subtracting. A small sign mistake can lead to the wrong answer.

So, why do we care about all this? Well, understanding function subtraction is a key concept in mathematics. It is used in so many areas, such as calculus, statistics, and even in computer science. Being able to manipulate and understand functions is a fundamental skill that opens the door to more complex math. It helps you see how different equations relate to each other. Furthermore, this skill is really important for problem-solving. This kind of thinking helps in real life, not just in math class! When you understand how functions work, you're better at solving problems in general.

Step-by-Step Solution: Finding the Value of x

Okay, now that we're all on the same page, let's solve this thing! We have f(x) = 16x - 30 and g(x) = 14x - 6. We want to find x such that (f - g)(x) = 0. Here’s what we do:

  1. Subtract the functions: (f - g)(x) = f(x) - g(x) = (16x - 30) - (14x - 6)

  2. Simplify the expression: 16x - 30 - 14x + 6 = 2x - 24

  3. Set the result equal to zero: 2x - 24 = 0

  4. Solve for x: 2x = 24 x = 12

Boom! We found it! The value of x that makes (f - g)(x) = 0 is 12.

So, as you can see, solving for x in a function subtraction problem is like following a recipe. Each step builds on the one before it. Let's make sure that everyone understands how we got to the solution. In the first step, we just wrote out the basic formula. Then we plugged in what the problem gave us for both f(x) and g(x). Be super careful with the negative signs! A common mistake is forgetting to distribute the negative sign to both terms inside the parentheses. So we end up with +6 instead of -6. Then we combined like terms to simplify the expression, getting 2x - 24. The next step is to set this new equation to zero. This step is based on the question that was asked. Then we solved for x. This involves a couple of simple steps: adding 24 to both sides, then dividing by 2. This is the goal – finding the value of x. Make sure to double-check that your answer makes sense. This helps catch any simple mistakes. Substituting x back into the original functions is a great way to verify that your answer is correct. Remember, the point of all this is to find where the two functions intersect (or where the subtracted function crosses the x-axis). When we look at the whole process, it’s not too complicated. The key is to take it one step at a time and not to get intimidated by the math. If you do this enough, it will become easy. Keep practicing, and it will become easier and easier. And don't be afraid to ask for help if you're stuck.

Analyzing the Answer Choices: Finding the Correct Solution

Let’s check the answer choices. We got x = 12. Looking at the options, we can see that C. 12 is the correct answer. The other choices are incorrect. Great job! Let's get more in-depth. We know the answer. Now, let’s go over why the other choices aren't correct. Understanding why the wrong answers are wrong is almost as important as getting the correct answer. This helps us learn and avoid similar mistakes in the future.

  • A. -18: This answer is incorrect. It suggests a possible mistake during the simplification or solving process. Perhaps a sign error? Always double-check your calculations. It is really easy to overlook a negative sign, so keep an eye out for that!
  • B. -12: Also incorrect. This could be the result of a calculation error. Go back and check your work. Maybe there was an error in combining like terms, or a miscalculation during the equation.
  • D. 18: Incorrect. This is very close! It’s possible that there was a minor mistake during solving the equation. The difference between 12 and 18 shows that one step of the work was missed or improperly calculated.

So, by carefully comparing our answer with the given options, we can be confident that 12 is the right solution. Looking back at each step of the process and verifying that we did everything correctly, is a must. If there's an exam, you have to be accurate. We can do that by plugging the answer back into the equation. It will confirm if everything checks out.

Why This Matters: Real-World Applications

Okay, you might be asking yourself,