Adding Functions: Find F(x) + G(x) Easily!

Hey math enthusiasts! Today, we're diving into a fun concept: adding functions. Specifically, we'll find the sum of two functions, f(x) and g(x). We are given the functions f(x) = 3.7 - 2x and g(x) = 0.25x - 5. Our goal is to calculate f(x) + g(x). Let's break it down step by step to make it super clear. Understanding this is a cornerstone for many advanced mathematical concepts, so pay close attention. It's like learning the building blocks of a Lego castle; once you get the basics, you can create anything!

Understanding the Basics of Function Addition

Adding functions might sound complex, but it's really straightforward. When we say f(x) + g(x), we simply mean we need to add the expressions for f(x) and g(x) together. Think of it like combining two different recipes to make a new dish. Each function is a recipe, and the sum is the combined dish. In our case, f(x) is the first recipe and g(x) is the second. We take the ingredients (the expressions) from both recipes and combine them. So, instead of being intimidated, view it as a fun puzzle that we need to solve together. The great thing about this is once you get the hang of it, you'll be able to add any kind of functions, whether they are simple linear equations or more complex polynomial functions. Believe me, it's easier than it sounds!

In our case, f(x) = 3.7 - 2x and g(x) = 0.25x - 5. To find f(x) + g(x), we'll add these two expressions. We are essentially merging these two separate functions into one. It will be a new function, let's call it h(x), which represents the sum of f(x) and g(x). This new function h(x) will have a unique relationship with the original x values.

Step-by-Step Calculation

Alright, let's get into the nitty-gritty and calculate f(x) + g(x). Here is the breakdown:

1. Write down the expressions: First, write down both functions. We have f(x) = 3.7 - 2x and g(x) = 0.25x - 5.
2. Add the functions: Now, add f(x) and g(x) together. This means we combine the terms in each expression: f(x) + g(x) = (3.7 - 2x) + (0.25x - 5).
3. Combine like terms: Next, group the like terms together. Like terms are those that have the same variable (in this case, x) or are constants (numbers without a variable). We have two types of like terms: constant numbers (3.7 and -5) and terms with x (-2x and 0.25x). Combine them: 3.7 - 5 - 2x + 0.25x.
4. Simplify: Finally, perform the addition and subtraction: 3.7 - 5 = -1.3 and -2x + 0.25x = -1.75x. Therefore, f(x) + g(x) = -1.3 - 1.75x.

So, after all the steps, the sum of f(x) and g(x) is -1.3 - 1.75x. Not so bad, right?

Detailed Explanation of Each Step

Let's delve deeper into each step. The goal here is not just to get the answer but to really grasp the method behind it. This understanding will help you tackle more complicated function problems in the future. We're going to break down each step so that every concept is understandable.

Writing Down the Expressions

This initial step is crucial because if you write the functions down incorrectly, all subsequent calculations will be off. Ensure that you correctly transcribe f(x) = 3.7 - 2x and g(x) = 0.25x - 5. Double-check your work to avoid any silly errors. Remember, accuracy is key, and we always want to start on the right foot.

Adding the Functions

Adding the functions means that you literally add the expressions. It’s important to understand the notation. f(x) + g(x) is the same as (3.7 - 2x) + (0.25x - 5). Notice how the entire expressions are included within parentheses. This is to ensure that the signs and terms are correctly carried over. This sets the groundwork for the next stage where the true arithmetic happens.

Combining Like Terms

Combining like terms is about grouping similar elements together. Here, it’s about identifying and grouping constants together (3.7 and -5) and terms with the variable x (-2x and 0.25x). The key concept is that you can only add or subtract terms if they are like terms. Think of it like adding apples to apples and oranges to oranges. You can’t directly add apples and oranges because they are different. Likewise, you can't combine a constant number with a term containing a variable.

Simplifying the Expression

Simplifying is the arithmetic phase where you perform the additions and subtractions. For example, 3.7 - 5 gives us -1.3, and -2x + 0.25x equals -1.75x. Each step here is a basic arithmetic operation. Careful attention to signs (positive or negative) is essential to get the correct answer. Remember that when you're combining terms with x, you only add or subtract the coefficients (the numbers in front of x).

Analyzing the Answer Choices

Now that we've found our answer, h(x) = -1.3 - 1.75x, let's compare it to the answer choices provided. This is where we confirm our work and ensure that we've chosen the correct one. The question provided multiple choices; let's see which option matches our calculations. This part is crucial for any test-taking scenario, as it helps you validate your answer.

Comparing with the Options

We need to compare our derived function with the answer choices. We've determined that f(x) + g(x) = -1.3 - 1.75x. Now, let’s look at the given options:

A. h(x) = 3.95x - 7 B. h(x) = 8.7 - 2.25x C. h(x) = -1.3 - 1.75x D. h(x) = 3.95 - 7x

Comparing our result with these options, we can see that Option C matches our calculated result perfectly. Therefore, Option C is the correct answer. This simple comparison step is a great way to verify your answer and boost your confidence in your calculations.

Why the Other Options are Incorrect

Let’s quickly address why the other options are wrong. Option A and Option D have incorrect coefficients for x and incorrect constant terms. Option B has incorrect coefficients for x and an incorrect constant term as well. Because we correctly calculated the sum of the functions, these options could not match our answer. Identifying the errors in the wrong answers helps you understand the mistakes to avoid. This process is just as important as getting the correct answer!

Common Mistakes and How to Avoid Them

Let's talk about some common pitfalls and how to steer clear of them. Recognizing and understanding these mistakes helps prevent them in the future. Remember, math is like a game; knowing the rules prevents you from making mistakes.

Sign Errors

One of the most frequent mistakes is sign errors. For instance, incorrectly handling the minus signs, like miscalculating -2x + 0.25x. Always double-check the signs and make sure you're adding and subtracting the terms correctly. Remember, a single incorrect sign can change your whole answer.

Incorrect Combination of Terms

Sometimes, you might try to combine unlike terms (e.g., adding a constant to a term with x). Make sure you're only combining like terms. This means only adding constants to constants and terms with x to terms with x. This error often comes from not properly understanding the order of operations and the rules of algebraic manipulation.

Forgetting to Distribute

If the functions involve more complex expressions, there might be a need to distribute (like in cases where there are parentheses). Double-check to see if you have performed all the necessary distribution steps correctly. This step ensures that all terms are handled properly.

Not Simplifying Completely

Sometimes, students find the like terms but then fail to simplify them fully. Always combine all the like terms to get to the simplest form. Failing to simplify completely means you haven't arrived at the final, correct answer. Always check and make sure that you've reduced the expression as much as possible.

Conclusion: Mastering Function Addition

Alright, folks, you've now successfully learned how to add two functions together! We've gone through the steps, understood the concepts, and identified the common mistakes to avoid. Remember, the key is to take it step by step, understand the underlying principles, and practice. Practice makes perfect, and with a bit of effort, you'll be adding functions like a pro in no time.

To recap: 1. Write down the expressions. 2. Add the functions together. 3. Combine the like terms. 4. Simplify. By following these steps and double-checking your work, you'll be well-prepared to tackle any function addition problem. Keep practicing, stay curious, and keep exploring the wonderful world of mathematics! You've got this!