Evaluating (f-g)(2) Given F(x) = 3x^2 + 1 And G(x) = 1 - X

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In the realm of mathematics, functions are fundamental building blocks that describe relationships between variables. Operations on functions, such as addition, subtraction, multiplication, and composition, allow us to combine and manipulate functions to create new ones. This article delves into the concept of function subtraction and demonstrates how to evaluate the difference of two functions at a specific point. We will focus on the specific example where f(x) = 3x^2 + 1 and g(x) = 1 - x, and our goal is to find the value of (f - g)(2). This problem is a classic example of how function operations work, and understanding it will help you tackle more complex problems in calculus and other areas of mathematics. Let's break down the process step by step, ensuring a clear and comprehensive understanding.

Defining Function Operations

Before we dive into the specific problem, let's clarify what we mean by function operations. Just like we can perform arithmetic operations on numbers, we can also perform operations on functions. These operations create new functions based on the original ones. The four basic operations are:

  • Addition: (f + g)(x) = f(x) + g(x)
  • Subtraction: (f - g)(x) = f(x) - g(x)
  • Multiplication: (f * g)(x) = f(x) * g(x)
  • Division: (f / g)(x) = f(x) / g(x), where g(x) ≠ 0

In our case, we are interested in function subtraction, which means we will be subtracting the function g(x) from the function f(x). This operation results in a new function, (f - g)(x), which is defined as the difference between the values of f(x) and g(x) for a given input x. Understanding this basic definition is crucial for solving the problem at hand. We must remember that the order of subtraction matters; (f - g)(x) is not necessarily the same as (g - f)(x). Now that we have a solid understanding of function subtraction, let's apply this concept to our specific functions.

Applying Function Subtraction to f(x) and g(x)

Now, let's apply the definition of function subtraction to our given functions, f(x) = 3x^2 + 1 and g(x) = 1 - x. To find (f - g)(x), we simply subtract g(x) from f(x):

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

Substitute the expressions for f(x) and g(x):

(f - g)(x) = (3x^2 + 1) - (1 - x)

Next, we need to simplify the expression by distributing the negative sign and combining like terms. This is a crucial step as it allows us to obtain a simplified form of the new function (f - g)(x). Accuracy in this step is paramount to avoid errors in the final result. Remember, distributing the negative sign correctly is key to obtaining the correct simplified expression. Let's proceed with the simplification:

(f - g)(x) = 3x^2 + 1 - 1 + x

Combine the constant terms (1 and -1):

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

So, the resulting function (f - g)(x) is 3x^2 + x. This new function represents the difference between the original functions f(x) and g(x). We have successfully performed the function subtraction and now have a simplified expression for (f - g)(x). The next step is to evaluate this new function at the specific point x = 2. This will give us the value of (f - g)(2), which is the ultimate goal of our problem.

Evaluating (f-g)(2)

We have determined that (f - g)(x) = 3x^2 + x. Now, we need to find the value of this function when x = 2. This is a straightforward substitution process. We replace every instance of x in the expression 3x^2 + x with the value 2. This will give us a numerical value for (f - g)(2). It's important to follow the order of operations (PEMDAS/BODMAS) to ensure we calculate the correct result. First, we'll square the 2, then multiply by 3, and finally add the result to 2. Let's perform the substitution:

(f - g)(2) = 3(2)^2 + 2

Now, follow the order of operations:

  • First, calculate the exponent: 2^2 = 4
  • Next, perform the multiplication: 3 * 4 = 12
  • Finally, add the terms: 12 + 2 = 14

Therefore, (f - g)(2) = 14. This is the final answer to our problem. We have successfully evaluated the difference of the two functions at the given point. This process demonstrates how function operations and evaluation work together to provide solutions in mathematics. Understanding these concepts is essential for further studies in calculus and related fields.

Alternative Approach: Evaluating f(2) and g(2) Separately

While we found (f - g)(x) first and then evaluated it at x = 2, there's another valid approach. We can evaluate f(2) and g(2) separately and then subtract the results. This method provides an alternative way to solve the problem and can sometimes be more convenient depending on the complexity of the functions. Let's explore this alternative approach.

First, let's evaluate f(2). Recall that f(x) = 3x^2 + 1. Substitute x = 2:

f(2) = 3(2)^2 + 1

Follow the order of operations:

  • 2^2 = 4
  • 3 * 4 = 12
  • 12 + 1 = 13

So, f(2) = 13.

Next, let's evaluate g(2). Recall that g(x) = 1 - x. Substitute x = 2:

g(2) = 1 - 2 = -1

Now, subtract g(2) from f(2):

(f - g)(2) = f(2) - g(2) = 13 - (-1) = 13 + 1 = 14

As we can see, we arrive at the same answer, (f - g)(2) = 14, using this alternative method. This confirms the correctness of our previous solution and highlights the flexibility in approaching function operation problems. Choosing the most efficient method often depends on the specific functions involved and personal preference. This alternative approach reinforces the understanding of function subtraction and evaluation.

Conclusion

In this article, we explored the concept of function subtraction and successfully evaluated (f - g)(2) given f(x) = 3x^2 + 1 and g(x) = 1 - x. We demonstrated two methods: first, finding the expression for (f - g)(x) and then substituting x = 2, and second, evaluating f(2) and g(2) separately and then subtracting the results. Both methods yielded the same answer, 14, reinforcing the understanding of function operations. This exercise provides a solid foundation for tackling more complex problems involving functions. Understanding function operations is crucial for success in calculus and other advanced mathematical topics. Remember to practice these concepts to build your proficiency and confidence in working with functions. The ability to manipulate and evaluate functions is a valuable skill in many areas of mathematics and science.