Evaluating (f-g)(144) Given F(x) = X^2 - 4 And G(x) = X + 2

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In the realm of mathematics, functions play a pivotal role in describing relationships and patterns. Function operations, such as subtraction, allow us to combine functions and explore their interactions. This article delves into the process of evaluating the composite function (f-g)(144), given the functions f(x) = x^2 - 4 and g(x) = x + 2. We will break down the steps involved, providing a comprehensive understanding of the underlying concepts and techniques.

Defining Function Operations: Subtraction

Before we dive into the specific problem, let's first understand the concept of function subtraction. When we subtract one function from another, we are essentially creating a new function whose output is the difference between the outputs of the original functions for a given input. Mathematically, we represent this as:

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

This means that to find the value of (f - g)(x) for a particular value of x, we first evaluate f(x) and g(x) separately and then subtract the value of g(x) from the value of f(x).

Step-by-Step Evaluation of (f-g)(144)

Now, let's apply this concept to our specific problem. We are given the functions f(x) = x^2 - 4 and g(x) = x + 2, and we want to find the value of (f - g)(144). Here's how we can do it step-by-step:

1. Find the Expression for (f-g)(x)

First, we need to find the general expression for (f - g)(x). Using the definition of function subtraction, we have:

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

To simplify this expression, we need to distribute the negative sign and combine like terms:

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

So, the expression for the composite function (f - g)(x) is x^2 - x - 6. This expression represents a new function that we can use to evaluate (f - g)(x) for any value of x.

2. Substitute x = 144 into the Expression

Now that we have the expression for (f - g)(x), we can substitute x = 144 to find the value of (f - g)(144):

(f - g)(144) = (144)^2 - 144 - 6

3. Calculate the Value

Next, we need to perform the calculations to find the numerical value of (f - g)(144). First, we calculate 144 squared:

(144)^2 = 144 * 144 = 20736

Now, we substitute this value back into the expression:

(f - g)(144) = 20736 - 144 - 6

Finally, we perform the subtraction:

(f - g)(144) = 20736 - 150 = 20586

Therefore, the value of (f - g)(144) is 20586.

Alternative Method: Evaluating f(144) and g(144) Separately

There's also an alternative method to solve this problem. Instead of finding the general expression for (f - g)(x) first, we can evaluate f(144) and g(144) separately and then subtract the results. Let's see how this works:

1. Evaluate f(144)

Substitute x = 144 into the expression for f(x):

f(144) = (144)^2 - 4 = 20736 - 4 = 20732

2. Evaluate g(144)

Substitute x = 144 into the expression for g(x):

g(144) = 144 + 2 = 146

3. Subtract g(144) from f(144)

Now, subtract the value of g(144) from the value of f(144):

(f - g)(144) = f(144) - g(144) = 20732 - 146 = 20586

As we can see, this method also gives us the same result: (f - g)(144) = 20586.

Key Concepts and Takeaways

This problem illustrates several important concepts related to functions and function operations:

  • Function Subtraction: Understanding how to subtract one function from another is crucial for working with composite functions.
  • Evaluating Functions: We need to be able to substitute values into functions and perform the necessary calculations.
  • Simplifying Expressions: Simplifying the expression for (f - g)(x) can make the evaluation process easier.
  • Alternative Methods: There are often multiple ways to solve a problem, and it's helpful to be aware of different approaches.

Common Mistakes to Avoid

When working with function operations, there are a few common mistakes that students often make. Here are some to watch out for:

  • Incorrectly Distributing the Negative Sign: When subtracting functions, it's important to distribute the negative sign to all terms in the second function. For example, (x^2 - 4) - (x + 2) should be simplified as x^2 - 4 - x - 2, not x^2 - 4 - x + 2.
  • Combining Unlike Terms: Only like terms can be combined. For example, x^2 and x are unlike terms and cannot be combined.
  • Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS) when performing calculations.

Practice Problems

To solidify your understanding of function operations, try these practice problems:

  1. If f(x) = 2x^2 + 3 and g(x) = x - 1, find (f - g)(2).
  2. If f(x) = sqrt(x) and g(x) = x + 5, find (f - g)(4).
  3. If f(x) = x^3 and g(x) = 2x, find (f - g)(-1).

By working through these problems, you'll gain confidence in your ability to evaluate composite functions and apply the concepts we've discussed.

Conclusion

In conclusion, evaluating (f - g)(144) given f(x) = x^2 - 4 and g(x) = x + 2 involves understanding the concept of function subtraction, simplifying expressions, and performing calculations. We explored two methods for solving this problem: finding the general expression for (f - g)(x) first and evaluating f(144) and g(144) separately. Both methods lead to the same answer: (f - g)(144) = 20586. By understanding the underlying concepts and practicing regularly, you can master function operations and excel in your mathematical pursuits. Remember to pay close attention to detail, avoid common mistakes, and explore different approaches to problem-solving. With dedication and practice, you can confidently tackle any function operation problem that comes your way.

This exploration of function subtraction and evaluation highlights the interconnectedness of mathematical concepts and the importance of a solid foundation in algebra. As you continue your mathematical journey, remember that practice and persistence are key to success. Embrace challenges, learn from your mistakes, and celebrate your accomplishments. With a growth mindset and a passion for learning, you can unlock the beauty and power of mathematics.