Finding (f-g)(x) Given F(x) And G(x)

Hey guys! Let's dive into a common type of math problem where we need to find the difference between two functions. Specifically, we're going to figure out how to calculate $(f-g)(x)$ when we're given the functions $f(x)$ and $g(x)$. This kind of problem pops up a lot in algebra and calculus, so understanding the steps is super important. We will break it down in a way that's easy to follow, so you can tackle these problems with confidence.

Understanding Function Subtraction

Before we jump into the example, let's make sure we're all on the same page about what $(f-g)(x)$ actually means. Function subtraction is exactly what it sounds like: we're subtracting one function from another. In this case, $(f-g)(x)$ means we're taking the function $f(x)$ and subtracting the function $g(x)$ from it. Mathematically, we can write this as:

$$(f-g)(x) = f(x) - g(x)$$

This simple equation is the key to solving these types of problems. We're just going to plug in the expressions for $f(x)$ and $g(x)$ and then simplify. It's like subtracting regular algebraic expressions, but with a fancy function notation. Keep this in mind as we move forward; it's the foundation for everything else we'll be doing. With this basic understanding, let's jump into our example problem and see how it works in practice. Remember, the goal is to make this concept crystal clear, so you can handle any function subtraction problem that comes your way.

Our Specific Problem: $f(x) = 2\sqrt{m}$ and $g(x) = 3\sqrt{m}$

Okay, now let's get to the heart of the matter. We're given two functions:

• $f(x) = 2\sqrt{m}$
• $g(x) = 3\sqrt{m}$

Our mission, should we choose to accept it (and we do!), is to find $(f-g)(x)$. Remember from our earlier discussion that $(f-g)(x)$ is the same as $f(x) - g(x)$. So, let's rewrite our problem using this understanding:

$$(f-g)(x) = 2\sqrt{m} - 3\sqrt{m}$$

Now we've transformed our function problem into a straightforward algebraic expression. Notice that both terms on the right side have the same radical part, $\sqrt{m}$. This is crucial because it means we can combine these terms. It's just like combining like terms in a regular algebraic expression, such as $2x - 3x$. The only difference here is that we have $\sqrt{m}$ instead of $x$. Keep in mind that $m$ is a variable inside the square root, so it behaves a bit differently than a simple variable outside a radical. However, the principle of combining like terms still applies. So, let's move on and actually combine these terms to find our answer.

Solving for $(f-g)(x)$

Now comes the fun part – the actual calculation! We've already established that:

$$(f-g)(x) = 2\sqrt{m} - 3\sqrt{m}$$

Since both terms have the common factor of $\sqrt{m}$, we can think of this as combining like terms. It's just like saying "2 of something minus 3 of the same thing." In this case, the "something" is $\sqrt{m}$. So, we simply subtract the coefficients (the numbers in front of the $\sqrt{m}$):

$$(2 - 3)\sqrt{m}$$

Now, what is 2 minus 3? It's -1, of course! So we have:

$$(-1)\sqrt{m}$$

We can write this in a slightly cleaner way by simply writing $-\sqrt{m}$. This means the same thing, but it looks a bit more elegant. So, we've found our answer:

$$(f-g)(x) = -\sqrt{m}$$

That's it! We've successfully calculated $(f-g)(x)$. Now, let's see how this answer lines up with the options given in the original problem.

Comparing Our Solution with the Given Options

Alright, we've found that $(f-g)(x) = -\sqrt{m}$. Now, let's take a look at the answer choices provided in the original problem and see which one matches our solution. The options were:

A. $(f-g)(x)=-4 \sqrt{m}$ B. $(f-g)(x)=11 \sqrt{m}$ C. $(f-g)(x)=-7 \sqrt{m}$ D. $(f-g)(x)=8 \sqrt{m}$

Comparing our answer, $-\sqrt{m}$, to the options, we can see that none of the provided options exactly match. Our answer is simply the negative square root of m, while the options all have different coefficients in front of the square root. This is a crucial step in problem-solving – always double-check your answer against the given choices. If none of them match, it's a sign that either there might be an error in the problem itself, or we might have made a mistake in our calculations. In this case, it seems like there might be a slight error in the original answer choices. Our calculated answer of $-\sqrt{m}$ is the correct one based on the given functions $f(x)$ and $g(x)$. So, even though it doesn't match the options, we're confident in our solution.

Key Takeaways and Practice

Great job, guys! You've just walked through how to find $(f-g)(x)$ given two functions, $f(x)$ and $g(x)$. Let's recap the key steps we took:

1. Understand the notation: $(f-g)(x)$ means $f(x) - g(x)$.
2. Substitute: Plug in the expressions for $f(x)$ and $g(x)$.
3. Combine like terms: Simplify the expression by combining terms with the same radical or variable part.
4. Double-check: Compare your answer to the given options (if any) and make sure it makes sense.

Remember, function subtraction is a fundamental concept, and with a little practice, you'll become super comfortable with it. Now, to really solidify your understanding, try working through some similar problems on your own. You can change the coefficients or even the expressions inside the square root to create new challenges. For instance, try finding $(f-g)(x)$ if $f(x) = 5\sqrt{m}$ and $g(x) = 2\sqrt{m}$, or even try it with different functions altogether, like $f(x) = x^2$ and $g(x) = x$. The more you practice, the more natural this process will become. Keep up the great work, and you'll be a function subtraction pro in no time!