Finding (f+g)(-7) For Given Functions F(x) And G(x)

Hey guys! Today, we're diving into a fun math problem involving functions. Specifically, we're going to figure out how to find the value of $(f+g)(-7)$ when we're given two functions: $F(x) = -6x + 17$ and $g(x) = 3x - 15$. This might sound a little intimidating at first, but trust me, it's totally manageable once we break it down step by step. We will go through each step thoroughly, ensuring that everyone can follow along and understand the concepts. So grab your pencils, and let's get started!

Understanding the Problem

So, what exactly are we trying to do here? Our main goal is to find the value of a new function, $(f+g)(x)$, at a specific point, which is $x = -7$. This new function, $(f+g)(x)$, is simply the sum of the two functions we're given, $F(x)$ and $g(x)$. In simpler terms, it means we need to add the expressions for $F(x)$ and $g(x)$ together. Think of it like combining two recipes to create a new dish! We need to understand the role of each function and how they interact when combined. This foundational understanding is crucial for solving the problem accurately and efficiently.

To really get a grip on this, let's break down the given functions. We have $F(x) = -6x + 17$. This is a linear function, meaning its graph would be a straight line. The $-6x$ part tells us that for every increase of 1 in $x$, the value of the function decreases by 6. The $+17$ is the y-intercept, where the line crosses the vertical axis. Understanding these components helps us visualize how the function behaves. Similarly, $g(x) = 3x - 15$ is also a linear function. Here, for every increase of 1 in $x$, the function value increases by 3, and the y-intercept is -15. Recognizing these patterns is key to solving not just this problem, but many others involving linear functions. This meticulous examination of the given functions forms the bedrock of our problem-solving strategy.

Now that we have a solid understanding of what the problem is asking and the nature of the functions involved, we can move on to the next step: actually adding the functions together. This involves a bit of algebraic manipulation, but don't worry, we'll take it nice and slow, making sure every step is clear. This is where the real fun begins – we get to put our knowledge into action and see how these functions combine! Remember, math isn't just about numbers; it's about understanding relationships and patterns. And that's exactly what we're doing here, uncovering the relationship between $F(x)$, $g(x)$, and their sum, $(f+g)(x)$.

Step-by-Step Solution

1. Find (f+g)(x)

Okay, so the first thing we need to do is figure out what $(f+g)(x)$ actually is. Remember, this just means we're adding the two functions together. So, we write it out like this:

$(f+g)(x) = F(x) + g(x)$

Now, let's substitute the expressions we have for $F(x)$ and $g(x)$:

$(f+g)(x) = (-6x + 17) + (3x - 15)$

Next up, we need to simplify this expression by combining like terms. Like terms are those that have the same variable raised to the same power. In this case, we have $-6x$ and $3x$, and we also have the constants $17$ and $-15$. Combining like terms is a fundamental algebraic skill, and it's crucial for simplifying expressions and solving equations. It allows us to work with more manageable expressions, making the rest of the problem-solving process much smoother. Think of it as decluttering your workspace before tackling a big project!

Let's start with the $x$ terms: $-6x + 3x = -3x$. This means that if you have -6 of something and you add 3 of it, you're left with -3. It's a basic arithmetic operation, but it's essential for the correct simplification. Now, let's move on to the constants: $17 - 15 = 2$. This is straightforward subtraction. So, when we combine the $x$ terms and the constants, we get:

$(f+g)(x) = -3x + 2$

Awesome! We've found the expression for $(f+g)(x)$. This new function represents the sum of our original two functions. It tells us how the combined function behaves for any value of $x$. Understanding how functions can be combined like this opens up a whole new world of possibilities in mathematics. Now that we have this combined function, we're ready to tackle the final step: plugging in $x = -7$ to find the value of $(f+g)(-7)$. This is where we'll see our hard work pay off, as we calculate the final answer to the problem.

2. Substitute x = -7

Now that we know $(f+g)(x) = -3x + 2$, we can find $(f+g)(-7)$ by simply substituting $-7$ for $x$ in the expression. This is a common technique in mathematics – once you have a general formula, you can find specific values by plugging in the relevant numbers. It's like having a recipe and then choosing the specific ingredients you want to use! This substitution process is a powerful tool, and it's used extensively in various mathematical contexts.

So, let's do it. We replace $x$ with $-7$ in our expression:

$(f+g)(-7) = -3(-7) + 2$

Now, we need to simplify this. Remember the order of operations (PEMDAS/BODMAS)? We need to do the multiplication before the addition. So, we multiply $-3$ by $-7$. A negative times a negative is a positive, so we get:

$-3(-7) = 21$

Now, we can substitute this back into our expression:

$(f+g)(-7) = 21 + 2$

Finally, we add $21$ and $2$:

$(f+g)(-7) = 23$

And there we have it! We've found the value of $(f+g)(-7)$. It's 23. This final step is the culmination of all our efforts, and it's satisfying to see how everything comes together to give us the answer. By following the steps carefully and methodically, we were able to solve the problem accurately and efficiently. This highlights the importance of breaking down complex problems into smaller, more manageable steps, a skill that's valuable not just in math, but in many aspects of life.

Final Answer

So, the final answer is:

$(f+g)(-7) = 23$

That means the correct answer is C) 23. We did it! We successfully found the value of $(f+g)(-7)$ by adding the functions $F(x)$ and $g(x)$ and then substituting $x = -7$. I hope this explanation was helpful and that you now feel more confident in tackling similar problems. Remember, the key is to break things down into smaller steps and to understand the underlying concepts. Keep practicing, and you'll become a function-solving pro in no time!

Key Takeaways

Let's recap some of the key concepts we covered in this problem. This is super important because understanding these concepts will help you tackle similar problems in the future. Think of these takeaways as the essential tools in your math toolbox!

• Function Notation: We saw how functions are represented using notation like $F(x)$ and $g(x)$. Understanding this notation is crucial for working with functions. It tells us that $F(x)$ is a function that takes $x$ as an input and produces a certain output. Similarly, $g(x)$ is another function with its own input and output. Getting comfortable with function notation is like learning the alphabet of mathematics – it's fundamental to understanding more complex ideas.
• Adding Functions: We learned how to add two functions together to create a new function, $(f+g)(x)$. This involves simply adding the expressions for the individual functions. This concept of combining functions is a powerful one, and it allows us to create new functions with different properties and behaviors. It's like combining ingredients in a recipe to create a new dish with a unique flavor.
• Combining Like Terms: We practiced combining like terms to simplify algebraic expressions. This is a fundamental skill in algebra, and it's essential for manipulating equations and expressions. Like terms are those that have the same variable raised to the same power. Combining them allows us to write expressions in a more concise and manageable form. Think of it as organizing your workspace – by grouping similar items together, you can make the whole space more efficient and effective.
• Substitution: We used substitution to find the value of $(f+g)(-7)$. This involves replacing the variable $x$ with a specific value, in this case, $-7$. Substitution is a widely used technique in mathematics, and it's essential for evaluating expressions and solving equations. It's like having a template and then filling in the blanks with specific information. This allows us to apply general formulas and rules to specific situations.
• Order of Operations: We followed the order of operations (PEMDAS/BODMAS) to simplify the expression. This ensures that we perform operations in the correct order, leading to the correct answer. The order of operations is a set of rules that dictates the sequence in which mathematical operations should be performed. It's like having a set of instructions for building something – following the instructions in the correct order is crucial for achieving the desired outcome.

By mastering these concepts, you'll be well-equipped to tackle a wide range of problems involving functions and algebraic expressions. Remember, math is like building a house – you need to lay a solid foundation of basic concepts before you can start constructing the more complex structures. So keep practicing, keep asking questions, and keep building your mathematical foundation!