Adding Functions: A Step-by-Step Guide

Hey math enthusiasts! Today, we're diving into the world of functions and how to add them together. Specifically, we'll be working through a problem where we need to find $(f+g)(x)$. This might sound a bit intimidating at first, but trust me, it's a piece of cake! We'll break down each step, making it super easy to understand. So, grab your pencils, and let's get started! We will go through the process step by step. This approach is great because it allows us to focus on the basics. By understanding the fundamental principles, we can tackle more complex function operations with ease. Understanding how to add functions is a fundamental concept in algebra, and it lays the groundwork for understanding more advanced topics like calculus. So, let's get this straight! If you're a bit rusty on functions, don't worry! We'll refresh the basics as we go along. The goal is to make sure everyone understands how to add functions, regardless of their current skill level. And remember, practice makes perfect! The more you work with functions, the more comfortable and confident you'll become. Let's get started!

Understanding the Basics: What are Functions?

Alright, before we jump into adding functions, let's quickly recap what a function is. Think of a function as a machine. You put something in (an input), and the machine does something to it (an operation), and then you get something out (an output). In math terms, a function takes an input value, often represented by x, and produces an output value, often represented by f(x) or g(x). For example, if we have the function $f(x) = x^2$, and we put in an input of 2, the function squares it, and the output is 4. It is pretty straightforward! The functions are mathematical relationships that map inputs to outputs. The notation $f(x)$ (pronounced "f of x") is used to denote the value of the function f at the input x. Essentially, the function f takes the input x, performs a specific operation, and produces an output. The operation can be any mathematical process, such as addition, subtraction, multiplication, division, exponentiation, or more complex combinations of these. Each function has a unique definition that dictates how it transforms the input into the output. The definition of a function provides the instructions or the algorithm that the function follows to calculate the output value. These function definitions are extremely helpful in more complex situations. They also provide a quick way to get the answer to the functions.

Now, let's consider the functions given in the problem: $f(x) = x^2 + 1$ and $g(x) = 5 - x$. The function $f(x)$ takes an input x, squares it, and then adds 1. The function $g(x)$ takes an input x and subtracts it from 5. These functions are different in their mathematical operation. We need to understand this to understand how to get the correct answer to the problem. This distinction is important because it highlights the independent nature of each function. Each function transforms an input in its specific way, following its rules to produce a distinct output. By understanding the unique operation of each function, we can more effectively combine them or analyze their behavior. We can create various combinations of functions, from addition and subtraction to multiplication and division, and even more complex compositions. Each combination produces a new function, offering a new mathematical relationship. The possibilities are endless. This capability allows us to model real-world phenomena, solve complex equations, and uncover hidden relationships in data. So, functions are like building blocks. It is important to be able to know all these functions and operations.

Finding $(f+g)(x)$: The Step-by-Step Solution

Alright, let's get to the main event: finding $(f+g)(x)$. The notation $(f+g)(x)$ simply means we need to add the two functions, $f(x)$ and $g(x)$, together. It's like saying, "What do we get if we combine these two function machines?" Here's how we do it, step by step:

1. Write down the functions: We have $f(x) = x^2 + 1$ and $g(x) = 5 - x$. It is very simple so far, right?

2. Add the functions: To find $(f+g)(x)$, we add the expressions for $f(x)$ and $g(x)$: $(f+g)(x) = f(x) + g(x)$ $(f+g)(x) = (x^2 + 1) + (5 - x)$ That's very easy.

3. Simplify the expression: Now, we just need to combine like terms. Notice that we have a constant of 1 and 5. Also, we have x and $x^2$. Let's write down the equation again. $(f+g)(x) = x^2 + 1 + 5 - x$ Combining like terms, we get: $(f+g)(x) = x^2 - x + 6$

So, the answer is $x^2 - x + 6$. Yay! That's it! We've successfully added the two functions and found $(f+g)(x)$. It is that easy to get the correct answer. The key is to understand the notation and the process. Don't worry if you don't understand it right away. You can always go over it again. The more you practice, the more comfortable you will be. Keep going, guys!

Matching the Answer with the Options

Now that we've found the solution, let's match it with the provided options to get the correct answer:

We found that $(f+g)(x) = x^2 - x + 6$. Let's go back to the options:

A. $x^2 + x - 4$ B. $x^2 + x + 4$ C. $x^2 - x + 6$ D. $x^2 + x + 6$

We can see that our answer, $x^2 - x + 6$, matches option C. Congratulations! You've successfully added the functions and selected the correct answer. So, the correct answer is C. The ability to add functions correctly is a foundational skill in algebra. This skill will be valuable as you continue your math journey.

Tips for Success

Here are some tips to help you master adding functions:

• Practice, practice, practice: The more problems you work through, the more comfortable you'll become. Try different variations of the problem! Change the functions. See if you can still get the answer. Get the basics right! Try to get the correct answer to a lot of functions. This will help you in your future math endeavors.
• Understand the notation: Make sure you understand what $(f+g)(x)$ means. Also, understand the difference between the functions.
• Pay attention to signs: Be very careful with positive and negative signs when simplifying expressions. It is important to be careful with the signs. One little mistake can throw you off and lead to an incorrect answer. This will cost you time, which is also valuable. So always pay attention.
• Break it down: If a problem seems complex, break it down into smaller steps. Start by writing the functions down. Then, add the functions. Then, simplify.
• Check your work: Always double-check your work to avoid careless errors. You can always go through your work again. This will help you get the correct answer. Double-check the signs as well.

Conclusion

Adding functions may seem difficult, but with practice, you can do it. Today, we walked through the process of adding two functions. Remember to take your time, understand each step, and practice. You've got this, guys!

Adding functions is a fundamental skill that opens the door to more complex mathematical concepts. By mastering this skill, you're building a strong foundation for future studies in algebra, calculus, and beyond. So keep practicing, keep learning, and never be afraid to ask questions. The world of mathematics is vast and exciting, and there's always more to discover! So have fun learning! The key is to be consistent and dedicated. Remember, every math problem you solve brings you closer to your goals. I hope this helps! Remember to ask questions if you need clarification. The more you understand, the more you can explore and uncover new mathematical adventures. So, good luck, and I hope this helps!