Evaluating Function Operations (g-f)(3) Where F(x) = 4 - X^2 And G(x) = 6x

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In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to corresponding outputs. Operations involving functions, such as addition, subtraction, multiplication, and composition, enable us to manipulate and combine these mathematical entities, revealing deeper relationships and patterns. This article delves into the realm of function operations, focusing on the subtraction of functions and the subsequent evaluation of the resulting expression at a specific point. We will specifically explore the scenario where f(x) = 4 - x^2 and g(x) = 6x, and our goal is to determine the expression equivalent to (g - f)(3). This exploration will not only solidify our understanding of function operations but also hone our skills in algebraic manipulation and evaluation.

The problem at hand presents us with two functions, f(x) and g(x), defined as f(x) = 4 - x^2 and g(x) = 6x. We are tasked with finding the expression equivalent to (g - f)(3). This notation signifies the subtraction of function f from function g, followed by the evaluation of the resulting function at x = 3. To embark on this journey, we must first grasp the concept of function subtraction. Function subtraction, denoted as (g - f)(x), involves subtracting the expression for f(x) from the expression for g(x). In essence, we are creating a new function that represents the difference between the outputs of g and f for a given input x. Once we have obtained the expression for (g - f)(x), we can then proceed to evaluate it at x = 3, which means substituting 3 for x in the expression and simplifying the result.

Understanding Function Subtraction

Before we dive into the specifics of our problem, let's take a moment to solidify our understanding of function subtraction. Function subtraction, as the name suggests, is the process of subtracting one function from another. Given two functions, f(x) and g(x), their difference, denoted as (g - f)(x), is defined as:

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

This definition is crucial for our task at hand. It tells us that to find the function (g - f)(x), we simply need to subtract the expression for f(x) from the expression for g(x). The resulting expression will be a new function that represents the difference between the outputs of g and f for any given input x. This concept is fundamental to understanding how functions interact and how we can manipulate them to extract new information.

In our specific case, we have f(x) = 4 - x^2 and g(x) = 6x. To find (g - f)(x), we will substitute these expressions into the definition of function subtraction:

(g - f)(x) = g(x) - f(x) = 6x - (4 - x^2)

Now, we need to simplify this expression by distributing the negative sign and combining like terms. This algebraic manipulation will give us a more concise and manageable expression for (g - f)(x). The ability to perform such algebraic manipulations is essential in working with functions, as it allows us to transform expressions into more convenient forms for analysis and evaluation.

Finding (g - f)(x)

Now that we have a solid understanding of function subtraction, let's apply this knowledge to our specific problem. We have f(x) = 4 - x^2 and g(x) = 6x, and we want to find (g - f)(x). Using the definition of function subtraction, we have:

(g - f)(x) = g(x) - f(x) = 6x - (4 - x^2)

The next step is to simplify this expression. We need to distribute the negative sign in front of the parentheses, which means changing the sign of each term inside the parentheses:

(g - f)(x) = 6x - 4 + x^2

Now, we can rearrange the terms to write the expression in a more standard form, with the term containing the highest power of x first:

(g - f)(x) = x^2 + 6x - 4

This is the expression for the function (g - f)(x). It represents the difference between the outputs of g(x) and f(x) for any given value of x. We have successfully found the function that results from subtracting f(x) from g(x). This is a crucial step towards our ultimate goal of evaluating (g - f)(3).

Now that we have the expression for (g - f)(x), we are ready to move on to the next step: evaluating this function at x = 3. This will involve substituting 3 for x in the expression and simplifying the result. The process of evaluating a function at a specific point is fundamental to understanding the function's behavior and its output for that particular input.

Evaluating (g - f)(3)

With the expression for (g - f)(x) in hand, we can now proceed to evaluate it at x = 3. This means substituting 3 for every instance of x in the expression:

(g - f)(3) = (3)^2 + 6(3) - 4

Now, we need to simplify this expression using the order of operations (PEMDAS/BODMAS). First, we evaluate the exponent:

(g - f)(3) = 9 + 6(3) - 4

Next, we perform the multiplication:

(g - f)(3) = 9 + 18 - 4

Finally, we perform the addition and subtraction from left to right:

(g - f)(3) = 27 - 4 = 23

Therefore, (g - f)(3) = 23. This is the value of the function (g - f)(x) when x = 3. We have successfully evaluated the function at the given point. This result tells us that the difference between the outputs of g(x) and f(x) when x = 3 is 23.

This completes the main part of our problem. We have found the expression for (g - f)(x) and evaluated it at x = 3. However, to fully address the problem, we need to compare our result with the given options and identify the equivalent expression. This will involve examining each option and determining whether it simplifies to the same value we obtained for (g - f)(3).

Comparing with the Options

Now, let's compare our result, (g - f)(3) = 23, with the given options to identify the equivalent expression.

  • Option A: 6 - 3 - (4 + 3)^2

    Let's simplify this expression:

    • 6 - 3 - (7)^2 = 6 - 3 - 49 = 3 - 49 = -46

    This is not equal to 23, so Option A is incorrect.

  • Option B: 6 - 3 - (4 - 3^2)

    Simplifying this expression:

    • 6 - 3 - (4 - 9) = 6 - 3 - (-5) = 6 - 3 + 5 = 3 + 5 = 8

    This is also not equal to 23, so Option B is incorrect.

  • Option C: 6(3) - 4 + 3^2

    Simplifying this expression:

    • 18 - 4 + 9 = 14 + 9 = 23

    This matches our result, so Option C is a potential answer.

  • Option D: 6(3) - 4 - 3^2

    Simplifying this expression:

    • 18 - 4 - 9 = 14 - 9 = 5

    This does not match our result, so Option D is incorrect.

Based on our analysis, Option C, 6(3) - 4 + 3^2, is the expression equivalent to (g - f)(3). We have systematically evaluated each option and compared it with our calculated value of 23, allowing us to confidently identify the correct answer.

Conclusion

In this article, we embarked on a journey to explore function operations, specifically the subtraction of functions, and the subsequent evaluation of the resulting expression at a specific point. We considered the functions f(x) = 4 - x^2 and g(x) = 6x, and our objective was to determine the expression equivalent to (g - f)(3).

We began by understanding the concept of function subtraction, which involves subtracting the expression for f(x) from the expression for g(x). We then applied this concept to our specific functions, finding the expression for (g - f)(x) as x^2 + 6x - 4. Subsequently, we evaluated this function at x = 3, obtaining the result (g - f)(3) = 23.

Finally, we compared our result with the given options, systematically simplifying each option and comparing it with our calculated value. This process led us to identify Option C, 6(3) - 4 + 3^2, as the expression equivalent to (g - f)(3).

This exploration not only solidified our understanding of function operations but also honed our skills in algebraic manipulation and evaluation. The ability to perform these operations is crucial in various areas of mathematics and its applications. By mastering these fundamental concepts, we pave the way for tackling more complex mathematical problems and gaining a deeper appreciation for the elegance and power of functions.