Function Operations: Solving (f+g)(-2), (f-g)(-2), And More

Hey guys! Today, we're diving deep into the fascinating world of function operations. Specifically, we're tackling a problem where we're given two functions, f(x) = 3x² + 12 and g(x) = x + 4, and we need to find the values of different operations on these functions at x = -2. This includes addition, subtraction, multiplication, and division. So, grab your calculators and let's get started!

Understanding Function Operations

Before we jump into the calculations, let's quickly recap what function operations actually mean. When we talk about (f + g)(x), (f - g)(x), (f * g)(x), and (f / g)(x), we're simply describing how to combine the outputs of the individual functions f(x) and g(x). Think of it like this: we're taking the results of applying each function to a particular input (in our case, -2) and then performing the indicated operation (addition, subtraction, multiplication, or division) on those results. This is a fundamental concept in algebra and calculus, and mastering it will open doors to more advanced topics. The beauty of function operations lies in their ability to create new functions from existing ones, allowing us to model complex relationships and solve a wider range of problems. For instance, in physics, we might use function operations to combine functions representing the position and velocity of an object to determine its acceleration. In economics, we could combine cost and revenue functions to calculate profit. Understanding these operations provides a powerful toolkit for analyzing and interpreting real-world phenomena. So, as we work through this example, remember that we're not just crunching numbers; we're building a foundation for understanding more complex mathematical concepts.

Addition of Functions: (f + g)(-2)

Let's begin with the addition of functions, (f + g)(-2). This means we need to find the sum of the values of f(x) and g(x) when x = -2. In mathematical terms, (f + g)(-2) = f(-2) + g(-2). So, our first step is to evaluate f(-2) and g(-2) separately.

For f(-2), we substitute -2 for x in the expression for f(x): f(-2) = 3(-2)² + 12. Remember the order of operations (PEMDAS/BODMAS): we first square -2, which gives us 4. Then, we multiply by 3, resulting in 12. Finally, we add 12, and we find that f(-2) = 12 + 12 = 24. It's crucial to follow the order of operations carefully to avoid making mistakes. Squaring the negative number first ensures that we get a positive result, which is essential for the rest of the calculation. This step highlights the importance of paying attention to detail in mathematical problem-solving. A small error in the initial stages can propagate through the entire solution, leading to an incorrect answer. So, always double-check your work, especially when dealing with exponents and negative numbers. By understanding the underlying principles and practicing these steps diligently, you'll build confidence and accuracy in your calculations. The ability to correctly evaluate functions at specific points is a fundamental skill that will serve you well in various mathematical contexts.

Now, let's evaluate g(-2). We substitute -2 for x in the expression for g(x): g(-2) = -2 + 4. This is a simple addition, and we find that g(-2) = 2. This part is relatively straightforward, but it's still important to get it right. Even a small mistake here can affect the final result. The function g(x) is a linear function, meaning its graph is a straight line. Evaluating it at a specific point simply involves substituting the value of x and performing the addition. This type of function is commonly encountered in various applications, such as modeling linear relationships between quantities. For example, it could represent the cost of a service based on the number of hours used. Understanding linear functions is crucial for building a strong foundation in algebra and calculus.

Finally, we add the results: (f + g)(-2) = f(-2) + g(-2) = 24 + 2 = 26. So, the value of (f + g)(-2) is 26. This completes the first part of our problem. We've successfully added the two functions at the given point. This process illustrates the simplicity and elegance of function operations. By breaking down the problem into smaller steps, we can systematically arrive at the solution. Remember, the key is to evaluate each function separately and then perform the indicated operation. This approach can be applied to any function operation, regardless of the complexity of the functions involved. The result, 26, represents the combined output of the two functions when x is -2. It's a single numerical value that captures the effect of adding the functions together at that specific point.

Subtraction of Functions: (f - g)(-2)

Next, we'll tackle the subtraction of functions, (f - g)(-2). This is similar to addition, but instead of adding the function values, we subtract them. In other words, (f - g)(-2) = f(-2) - g(-2). We already know that f(-2) = 24 and g(-2) = 2, so we can simply substitute these values into the expression.

Therefore, (f - g)(-2) = 24 - 2 = 22. The value of (f - g)(-2) is 22. Subtraction of functions is a straightforward process once you understand the concept. It's essential to pay attention to the order of subtraction, as f(x) - g(x) is generally not the same as g(x) - f(x). This highlights the importance of understanding the properties of mathematical operations. Subtraction is not commutative, meaning the order matters. In the context of functions, this means that the difference between two functions depends on which function is subtracted from the other. This can have significant implications in various applications. For instance, if f(x) represents the revenue of a company and g(x) represents its costs, then f(x) - g(x) represents the profit, while g(x) - f(x) would represent the loss. So, understanding the order of subtraction is crucial for interpreting the results correctly. The result, 22, represents the difference between the outputs of the two functions when x is -2. It tells us how much greater the value of f(x) is compared to g(x) at that specific point.

Multiplication of Functions: (f * g)(-2)

Now, let's move on to the multiplication of functions, (f * g)(-2). This means we multiply the values of f(x) and g(x) when x = -2. Mathematically, (f * g)(-2) = f(-2) * g(-2). Again, we know that f(-2) = 24 and g(-2) = 2.

So, (f * g)(-2) = 24 * 2 = 48. The value of (f * g)(-2) is 48. Multiplication of functions is another fundamental operation that allows us to combine functions in interesting ways. It's important to remember that when we multiply functions, we're multiplying their outputs, not their inputs. This is a key distinction that helps to understand the behavior of the resulting function. For example, if f(x) represents the length of a rectangle and g(x) represents its width, then (f * g)(x) represents the area of the rectangle. Multiplication can also be used to model exponential growth or decay. Understanding multiplication of functions is essential for building more complex mathematical models. The result, 48, represents the product of the outputs of the two functions when x is -2. It tells us how the two functions interact multiplicatively at that specific point. This can provide insights into the relationship between the functions and their overall behavior.

Division of Functions: (f / g)(-2)

Finally, let's consider the division of functions, (f / g)(-2). This means we divide the value of f(x) by the value of g(x) when x = -2. Mathematically, (f / g)(-2) = f(-2) / g(-2). Once more, we know that f(-2) = 24 and g(-2) = 2.

Therefore, (f / g)(-2) = 24 / 2 = 12. The value of (f / g)(-2) is 12. Division of functions introduces a crucial consideration: we need to ensure that the denominator, g(x), is not zero. If g(x) were zero, the division would be undefined. This highlights the importance of considering the domain of the resulting function when dividing functions. The domain is the set of all possible input values for which the function is defined. In the case of division, we need to exclude any values of x that would make the denominator zero. In our example, g(x) = x + 4, so g(x) would be zero when x = -4. Therefore, -4 is not in the domain of the function (f / g)(x). This concept is essential for understanding the behavior of functions and avoiding errors in calculations. The result, 12, represents the ratio of the outputs of the two functions when x is -2. It tells us how many times greater the value of f(x) is compared to g(x) at that specific point.

Conclusion

Alright guys, we've successfully navigated through all four function operations: addition, subtraction, multiplication, and division! We found that (f + g)(-2) = 26, (f - g)(-2) = 22, (f * g)(-2) = 48, and (f / g)(-2) = 12. This exercise demonstrates how we can combine functions to create new functions and evaluate them at specific points. Understanding function operations is crucial for success in higher-level mathematics, so keep practicing! Remember, the key to mastering these concepts is to break down the problem into smaller steps and understand the underlying principles. By carefully evaluating each function separately and then performing the indicated operation, you can confidently solve any function operation problem. So, keep up the great work, and you'll be a function operation pro in no time!