Subtracting Functions: A Step-by-Step Guide

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Hey guys! Let's dive into a fun problem involving function subtraction. We're given two functions, f(x) = -5^x - 4 and g(x) = -3x - 2, and our mission is to find (f-g)(x). Don't worry, it's not as intimidating as it looks! We'll break it down step by step, making sure everyone understands the process. Think of functions like little machines that take an input (in this case, x) and produce an output based on a specific rule. When we subtract functions, we're essentially subtracting the outputs of those machines for the same input. Let's get started!

Breaking Down Function Subtraction

The key to understanding (f-g)(x) lies in recognizing what it represents. (f-g)(x) simply means we're subtracting the function g(x) from the function f(x). Mathematically, we can write this as:

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

This is the fundamental concept we need to grasp. We're not multiplying or dividing; we're just taking the expression for f(x) and subtracting the entire expression for g(x) from it. It's like saying, "What's the difference between the output of f(x) and the output of g(x) for a given x?"

Now, let's apply this to our specific functions. We know that f(x) = -5^x - 4 and g(x) = -3x - 2. So, we can substitute these expressions into our equation:

(f-g)(x) = (-5^x - 4) - (-3x - 2)

See? We've simply replaced f(x) and g(x) with their respective formulas. The next step is where we need to be careful with our signs. We're subtracting the entire quantity of g(x), so we need to distribute the negative sign.

The Importance of Distributing the Negative Sign

This is a crucial step where many people can make mistakes, so pay close attention! We have a negative sign in front of the parentheses containing g(x). This means we need to distribute that negative sign to both terms inside the parentheses. It's like we're multiplying the entire expression (-3x - 2) by -1.

Let's rewrite our equation, distributing the negative sign:

(f-g)(x) = -5^x - 4 + 3x + 2

Notice what happened? The -3x became +3x, and the -2 became +2. This is because a negative times a negative is a positive. If we forget to distribute the negative sign correctly, we'll end up with the wrong answer. So, always double-check this step!

Now that we've successfully distributed the negative sign, we're ready to simplify our expression by combining like terms. This will give us our final answer for (f-g)(x).

Combining Like Terms for the Final Solution

In our expression, we have a constant term (-4) and another constant term (+2). These are like terms, meaning they can be combined. We also have a term with x (+3x) and a term with an exponent (-5^x). These are not like terms and cannot be combined.

So, let's combine the constant terms:

-4 + 2 = -2

Now, we can rewrite our equation with the simplified constant term:

(f-g)(x) = -5^x + 3x - 2

And there we have it! We've successfully found (f-g)(x). It's equal to -5^x + 3x - 2. This matches option A in our original question.

Therefore, the correct answer is:

A. (f-g)(x) = -5^x + 3x - 2

We made it! By carefully following the steps of function subtraction and paying close attention to the distribution of the negative sign, we arrived at the correct answer. Let's recap the key steps to make sure we've got this down.

Recap: Key Steps for Function Subtraction

To successfully subtract functions, remember these key steps:

  1. Understand the notation: (f-g)(x) = f(x) - g(x). This is the foundation of function subtraction.
  2. Substitute the function expressions: Replace f(x) and g(x) with their given formulas.
  3. Distribute the negative sign: This is the most crucial step! Make sure to multiply every term in g(x) by -1.
  4. Combine like terms: Simplify the expression by adding or subtracting terms that have the same variable and exponent.
  5. Double-check your work: It's always a good idea to go back and make sure you haven't made any mistakes, especially with the signs.

By following these steps, you'll be able to tackle any function subtraction problem with confidence. Now, let's try another example to solidify our understanding.

Practice Makes Perfect: A Second Example

Let's say we have two new functions:

  • h(x) = 2x^2 + 5x - 3
  • k(x) = x^2 - 2x + 1

And we want to find (h-k)(x). Can you guess the first step? That's right, it's substituting the function expressions:

(h-k)(x) = (2x^2 + 5x - 3) - (x^2 - 2x + 1)

Now, the all-important step: distributing the negative sign. Remember, we need to multiply each term inside the second parentheses by -1:

(h-k)(x) = 2x^2 + 5x - 3 - x^2 + 2x - 1

Notice how the signs of the terms in k(x) have changed. Now, let's combine like terms. We have 2x^2 and -x^2, 5x and 2x, and -3 and -1:

  • 2x^2 - x^2 = x^2
  • 5x + 2x = 7x
  • -3 - 1 = -4

So, our final answer is:

(h-k)(x) = x^2 + 7x - 4

See? Once you get the hang of it, function subtraction becomes quite straightforward. The key is to be methodical and pay attention to detail, especially when distributing the negative sign.

Why is Function Subtraction Important?

You might be wondering, "Okay, this is interesting, but why do we even need to learn about function subtraction?" Well, function subtraction has applications in various fields, including:

  • Calculus: Function subtraction is used in finding the difference between two curves, which is essential in calculating areas and other important concepts.
  • Physics: In physics, function subtraction can be used to find the net force acting on an object when multiple forces are involved.
  • Economics: Economists use function subtraction to analyze profit, which is the difference between revenue and cost functions.
  • Computer Graphics: In computer graphics, function subtraction can be used to create shadows and other visual effects.

So, understanding function subtraction is not just about solving math problems; it's about building a foundation for understanding more complex concepts in various disciplines. Plus, it's a great exercise for your brain!

Conclusion: You've Mastered Function Subtraction!

Great job, guys! You've successfully learned how to subtract functions. Remember the key steps: understand the notation, substitute the expressions, distribute the negative sign, combine like terms, and double-check your work. With practice, you'll become a function subtraction pro!

So, next time you encounter (f-g)(x), don't panic! You now have the tools and knowledge to tackle it with confidence. Keep practicing, and you'll be amazed at how much you can achieve in the world of mathematics. Keep exploring and have fun with functions!