Adding Functions: Step-by-Step Guide To Finding (f+g)(x)
Hey guys! Let's dive into a fun math problem today where we'll be exploring function addition. We're given two functions, $f(x) = 3x + 5$ and $g(x) = -2x^2 - 5x + 3$, and our mission is to find $(f+g)(x)$. Sounds like a quest, right? Don't worry, it's easier than you think! We'll break it down step by step, making sure everyone can follow along. So, buckle up and let's get started!
Understanding Function Addition
Before we jump into the specific problem, let's quickly recap what function addition actually means. When we see $(f+g)(x)$, it simply means we're adding the two functions, $f(x)$ and $g(x)$, together. Think of it like combining two ingredients in a recipe – you're just putting them together. The formal definition of the sum of two functions, denoted as $(f + g)(x)$, is defined as the sum of the functions $f(x)$ and $g(x)$ for every x in the domains of both f and g. This can be mathematically expressed as: $(f + g)(x) = f(x) + g(x)$. This operation is fundamental in various mathematical contexts and is used to model and solve a wide range of problems. For example, in physics, you might use function addition to combine the effects of different forces acting on an object. In economics, it could represent the total cost or revenue when combining different production processes. The key idea is that you're taking the output of each function for a given input x and adding those outputs together. This allows for a combined analysis or representation that incorporates the behavior of both functions simultaneously. Function addition is a versatile tool that enhances our ability to understand and manipulate mathematical relationships, making it an essential concept in both theoretical and applied mathematics. It’s like having two different perspectives on the same situation and then merging them to get a more complete picture. So, with this understanding, we can confidently move forward and tackle our problem.
Step 1: Write Down the Functions
Okay, first things first, let's write down the functions we're working with. This will help us keep things organized and prevent any mix-ups. We have:
See? Nothing scary here! We've just clearly stated what our functions are. Writing down the functions explicitly is a crucial first step in solving the problem. It allows us to have a clear reference point and avoids any confusion as we proceed with the addition. By clearly stating the functions, we lay the foundation for a structured and organized approach to the problem. This practice is particularly helpful when dealing with more complex functions or multiple operations. Ensuring we have the functions correctly written down at the start minimizes the chances of errors later on and streamlines the solution process. Moreover, this step emphasizes the importance of precision and attention to detail in mathematical problem-solving, reinforcing the idea that a careful setup is key to accurate results. It's like having the recipe ingredients laid out before you start cooking – it makes the whole process smoother and more efficient. So, with our functions neatly written down, we're ready to move on to the next step with confidence.
Step 2: Apply the Definition of Function Addition
Now for the fun part! Remember, $(f+g)(x)$ means we add $f(x)$ and $g(x)$. So, let's write that out:
This is just a symbolic representation of what we're about to do. It's like a roadmap telling us where we're going. The definition of function addition provides the framework for combining the two functions. This step is essential as it translates the symbolic notation into a concrete operation, guiding us towards the actual process of adding the functions together. By explicitly stating the definition, we solidify our understanding of what needs to be done and ensure we're on the right track. This step also highlights the elegance and precision of mathematical language, where symbols and definitions work together to convey complex ideas in a concise manner. It's like having a universal translator that allows us to express mathematical concepts clearly and unambiguously. So, with the definition in place, we're ready to substitute the actual expressions for $f(x)$ and $g(x)$ and move closer to our final answer. This step sets the stage for the algebraic manipulation that will follow, bringing us one step closer to unveiling $(f+g)(x)$.
Step 3: Substitute the Function Expressions
Time to substitute! We'll replace $f(x)$ and $g(x)$ with their actual expressions:
See how we just swapped the function names with what they actually equal? This is a crucial step in simplifying the expression. Substituting the function expressions is a fundamental step in evaluating $(f+g)(x)$. It bridges the gap between the abstract definition of function addition and the concrete algebraic manipulation required to obtain the result. By substituting the expressions, we transform the problem into a purely algebraic one, which we can then solve using standard techniques. This step emphasizes the importance of precision and accuracy in mathematics, as a correct substitution is essential for obtaining the correct final answer. It's like fitting the right pieces of a puzzle together – each piece has its place, and when they're all correctly positioned, the picture becomes clear. Furthermore, this step showcases the power of algebraic notation in representing mathematical relationships, enabling us to manipulate and simplify expressions efficiently. So, with the substitutions made, we're now poised to combine like terms and simplify the expression, bringing us closer to the final form of $(f+g)(x)$.
Step 4: Combine Like Terms
Now comes the algebra magic! We need to combine the terms that are similar. Remember, we can only add terms that have the same variable and exponent. Let's rewrite the expression without the parentheses to make it clearer:
Now, let's group the like terms together:
And finally, let's combine them:
Ta-da! We've done it! Combining like terms is a critical step in simplifying algebraic expressions, and it's where our algebraic skills really shine. This step involves identifying terms with the same variable and exponent and then adding or subtracting their coefficients. It's like sorting a collection of objects into groups of similar items – you're organizing the expression to make it more manageable. The process of combining like terms not only simplifies the expression but also reveals its underlying structure, making it easier to understand and interpret. This step demonstrates the power of algebraic manipulation in transforming complex expressions into simpler, more elegant forms. It's like refining a rough gem to reveal its brilliance. Moreover, it reinforces the importance of careful attention to signs and coefficients, ensuring that we combine the terms accurately. So, with the like terms combined, we've arrived at the simplified form of $(f+g)(x)$, which represents the sum of the two functions.
Step 5: Choose the Correct Answer
Alright, we've found that $(f+g)(x) = -2x^2 - 2x + 8$. Now, let's look at our options:
A. $(f+g)(x) = -2x^2 - 2x + 8$ B. $(f+g)(x) = -2x^2 + 8x + 8$ C. $(f+g)(x) = -4x + 8$ D. $(f+g)(x) = -2x + 8$
It looks like option A matches our result! So, that's our answer. Choosing the correct answer from the given options is the final step in the problem-solving process. This step requires careful comparison of our derived solution with the provided choices to identify the match. It's like finding the right key to unlock a door – you need to make sure the key fits the lock perfectly. This step reinforces the importance of accuracy and attention to detail throughout the entire solution process, as a small mistake earlier can lead to an incorrect final answer. It also highlights the value of checking our work and ensuring that our solution aligns with the given options. Moreover, this step provides a sense of closure and accomplishment, marking the successful completion of the problem. So, with confidence, we select the correct answer, knowing that we've followed a logical and systematic approach to arrive at the solution.
Conclusion: We Did It!
Awesome! We successfully found $(f+g)(x)$ by adding the two functions together. Remember, the key steps were:
- Write down the functions.
- Apply the definition of function addition.
- Substitute the function expressions.
- Combine like terms.
- Choose the correct answer.
By following these steps, you can tackle any function addition problem like a pro. Keep practicing, and you'll become a math whiz in no time! We did it! Concluding the problem is an important step that provides a sense of accomplishment and reinforces the key concepts learned. By summarizing the steps taken to solve the problem, we solidify our understanding of the process and create a valuable reference for future problem-solving. This step also allows us to reflect on the solution and identify any areas where we might need further practice or clarification. It's like reviewing the map after a journey – you can trace your route and identify the key landmarks along the way. Moreover, the conclusion provides an opportunity to encourage further exploration and practice, motivating us to continue learning and developing our mathematical skills. So, with a feeling of success and a clear understanding of the steps involved, we conclude the problem, ready to tackle new challenges and expand our mathematical horizons.
