Adding And Subtracting Functions: Find (f+g)(x) & (f-g)(x)

Hey guys! Today, we're diving into the world of function operations, specifically addition and subtraction. We'll tackle a problem where we need to find $(f+g)(x)$ and $(f-g)(x)$ given two functions, $f(x)$ and $g(x)$. Don't worry, it's not as intimidating as it sounds! We'll break it down step by step, so you'll be a pro in no time. So, let's jump right in and get started!

Understanding Function Operations

Before we dive into the specific problem, let's quickly recap what function addition and subtraction mean. When we talk about $(f+g)(x)$, we're simply adding the two functions $f(x)$ and $g(x)$ together. Similarly, $(f-g)(x)$ means we're subtracting the function $g(x)$ from $f(x)$. It's like combining like terms, but with functions! Understanding this basic concept is crucial for solving problems like this. Think of functions as little machines that take an input ($x$) and produce an output. Adding or subtracting functions is just a way of combining these machines to create a new one.

Function addition and subtraction are fundamental operations in mathematics, especially in calculus and analysis. They allow us to combine simpler functions to create more complex ones, or to break down complex functions into simpler components. This is particularly useful in modeling real-world phenomena, where a system's behavior might be described by a combination of different functions. For example, the total cost of a product might be modeled as the sum of a fixed cost function and a variable cost function. Similarly, the profit of a company can be modeled as the difference between the revenue function and the cost function. Mastering these operations provides a solid foundation for understanding more advanced mathematical concepts and their applications.

Furthermore, function operations extend beyond simple addition and subtraction. We can also perform multiplication, division, and composition of functions. Function composition, denoted as $(f ext{ o } g)(x)$ or $f(g(x))$, involves plugging one function into another. This operation is particularly important in understanding transformations of functions and is widely used in areas such as computer graphics and signal processing. Understanding all these function operations collectively empowers you to manipulate and analyze functions effectively, opening doors to a wide range of mathematical and scientific applications. So, while we're focusing on addition and subtraction in this example, remember that these are just the tip of the iceberg in the world of function operations!

Problem Setup: $f(x)$ and $g(x)$

Okay, let's get back to our problem. We're given two functions:

• $f(x) = 6x^2 + 5$
• $g(x) = 7 - 5x$

Our mission, should we choose to accept it (and we do!), is to find $(f+g)(x)$ and $(f-g)(x)$. So, basically, we need to add these functions together and then subtract them. This is where our understanding of function operations comes into play. We'll treat each function as a single entity and perform the operations as indicated. Remember, the key is to combine like terms, just like in regular algebra. The $x^2$ terms will go together, the $x$ terms will go together, and the constant terms will go together. Keep this in mind as we proceed with the calculations. Let's start with the addition part first. It's usually a bit more straightforward, and getting it right will build our confidence for the subtraction, where we need to be extra careful with the signs. Ready? Let's do it!

The expressions for $f(x)$ and $g(x)$ are polynomials, which are algebraic expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Polynomials are ubiquitous in mathematics and its applications, serving as fundamental building blocks for modeling various phenomena. The function $f(x) = 6x^2 + 5$ is a quadratic polynomial, characterized by the highest power of $x$ being 2. Quadratic functions are well-known for their parabolic graphs and are used to model projectile motion, optimization problems, and many other scenarios. On the other hand, $g(x) = 7 - 5x$ is a linear polynomial, as the highest power of $x$ is 1. Linear functions have straight-line graphs and are used to represent constant rates of change, direct proportionality, and simple relationships between variables. Recognizing the types of polynomials we're dealing with can provide insights into their behavior and properties, making it easier to manipulate and interpret them. In this case, knowing that we're adding and subtracting polynomials helps us anticipate that the result will also be a polynomial, and we can use our knowledge of polynomial arithmetic to efficiently perform the operations.

Finding $(f+g)(x)$

To find $(f+g)(x)$, we simply add the two functions:

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

Now, substitute the expressions for $f(x)$ and $g(x)$:

$(f+g)(x) = (6x^2 + 5) + (7 - 5x)$

Next, we combine like terms. Remember, like terms are those that have the same variable raised to the same power. In this case, we have a $6x^2$ term, a $-5x$ term, and two constant terms (5 and 7). So, let's rearrange the expression to group the like terms together:

$(f+g)(x) = 6x^2 - 5x + 5 + 7$

Finally, we add the constant terms:

$(f+g)(x) = 6x^2 - 5x + 12$

And there you have it! We've successfully found $(f+g)(x)$. See? It wasn't so bad after all. We just added the two functions and simplified the result by combining like terms. Now, let's move on to the subtraction part. This is where things can get a little trickier because we need to be careful with the negative sign. But with a little attention to detail, we'll nail it. We are really making progress here, guys!

When adding functions, the commutative and associative properties of addition hold true. The commutative property states that the order in which we add the functions does not matter, i.e., $f(x) + g(x) = g(x) + f(x)$. The associative property states that when adding three or more functions, the grouping of the functions does not affect the result, i.e., $(f(x) + g(x)) + h(x) = f(x) + (g(x) + h(x))$. These properties are not only fundamental in arithmetic but also play a crucial role in simplifying and manipulating function expressions. In this example, we implicitly used the commutative property when rearranging the terms to group like terms together. By understanding these properties, we can confidently rearrange and combine functions in various ways without altering the final result. This flexibility is particularly useful when dealing with more complex expressions or when trying to simplify an expression to a more manageable form. So, keep these properties in mind as you work with function operations; they're your allies in the world of mathematics!

Finding $(f-g)(x)$

Now, let's find $(f-g)(x)$. This means we need to subtract $g(x)$ from $f(x)$:

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

Again, we substitute the expressions for $f(x)$ and $g(x)$:

$(f-g)(x) = (6x^2 + 5) - (7 - 5x)$

Here's the tricky part! We need to distribute the negative sign to both terms inside the parentheses of $g(x)$. This is a common mistake people make, so pay close attention:

$(f-g)(x) = 6x^2 + 5 - 7 + 5x$

Notice how the $-7$ became $-7$ and the $-5x$ became $+5x$ after distributing the negative sign. This is super important! If you miss this step, you'll get the wrong answer. Now, we combine like terms as before:

$(f-g)(x) = 6x^2 + 5x + 5 - 7$

Finally, add the constant terms:

$(f-g)(x) = 6x^2 + 5x - 2$

Woohoo! We did it! We've found $(f-g)(x)$. See, even with the subtraction, we managed to get the correct answer by being careful with the signs. It's all about paying attention to those little details. Now that we've found both $(f+g)(x)$ and $(f-g)(x)$, we can confidently say we've conquered this problem. Let's take a moment to appreciate our hard work and then summarize our findings.

The distributive property is a fundamental concept in algebra that allows us to multiply a single term by two or more terms inside a set of parentheses. In the context of subtracting functions, the distributive property is crucial because we need to distribute the negative sign (which is like multiplying by -1) to every term in the function being subtracted. Failing to do so can lead to incorrect results, as we saw in the example. The distributive property can be expressed as $a(b + c) = ab + ac$, where $a$, $b$, and $c$ are any algebraic expressions. In our case, we had $-(7 - 5x)$, which is equivalent to $-1(7 - 5x)$. Applying the distributive property, we get $-1 * 7 + (-1) * (-5x) = -7 + 5x$, which is exactly what we did in the solution. Understanding and applying the distributive property correctly is essential for simplifying algebraic expressions and solving equations, making it a cornerstone of algebraic manipulation. So, always remember to distribute that negative sign when subtracting functions – it's a lifesaver!

Solution and Summary

So, let's recap our solutions:

• $(f+g)(x) = 6x^2 - 5x + 12$
• $(f-g)(x) = 6x^2 + 5x - 2$

We successfully found both $(f+g)(x)$ and $(f-g)(x)$ by carefully applying the definitions of function addition and subtraction. We combined like terms and paid close attention to the signs, especially when subtracting the functions. Remember, the key to success in these types of problems is to break them down into smaller steps, be mindful of the details, and double-check your work. Function operations might seem a little abstract at first, but with practice, they become second nature. And the more comfortable you are with them, the easier it will be to tackle more complex mathematical concepts down the road. You've done a great job following along, guys, and you're well on your way to mastering function operations!

In summary, function addition and subtraction involve combining two or more functions using the basic arithmetic operations of addition and subtraction. The result is a new function that represents the sum or difference of the original functions. To perform these operations, we simply add or subtract the corresponding expressions for the functions, being careful to distribute negative signs when subtracting. Then, we combine like terms to simplify the result. These operations are fundamental in calculus, analysis, and various other branches of mathematics, allowing us to manipulate and analyze functions effectively. By mastering function addition and subtraction, you'll gain a valuable tool for solving a wide range of mathematical problems and understanding real-world applications.

Practice Makes Perfect

Now that we've worked through this example together, the best way to solidify your understanding is to practice! Try working through similar problems on your own. You can find plenty of examples in textbooks, online resources, or even create your own! The more you practice, the more comfortable you'll become with function operations, and the faster you'll be able to solve them. Remember, math is like a muscle – the more you use it, the stronger it gets. So, don't be afraid to challenge yourself and keep practicing. You've got this! And if you ever get stuck, don't hesitate to ask for help. There are plenty of resources available, including your teachers, classmates, and online communities. Keep up the great work, and you'll be a function operation master in no time! You are amazing!