Evaluating (fg)(-2) Given F(x) = 8 - 10x And G(x) = 5x + 4

In this article, we will delve into the process of finding the value of (fg)(-2) given the functions f(x) = 8 - 10x and g(x) = 5x + 4. This problem involves understanding function composition and evaluation, fundamental concepts in mathematics. We will break down the steps involved, providing a clear and concise explanation to help you grasp the underlying principles.

Understanding Function Composition and Evaluation

Before we dive into the specific problem, let's briefly review the concepts of function composition and evaluation. Function composition refers to the process of combining two functions by applying one function to the result of the other. In the notation (fg)(x), it signifies the product of the functions f(x) and g(x). On the other hand, function evaluation involves substituting a specific value for the variable x in a function and calculating the resulting output.

In this case, we are given two functions: f(x) = 8 - 10x and g(x) = 5x + 4. Our goal is to find the value of (fg)(-2), which means we need to find the product of the functions f(x) and g(x) and then evaluate the resulting expression at x = -2.

Step-by-Step Solution

Let's break down the solution into manageable steps:

Step 1: Find the expression for (fg)(x)

To find (fg)(x), we simply multiply the expressions for f(x) and g(x):

(fg)(x) = f(x) * g(x) = (8 - 10x)(5x + 4)

Now, we need to expand this expression by using the distributive property (also known as the FOIL method):

(fg)(x) = 8(5x + 4) - 10x(5x + 4)

(fg)(x) = 40x + 32 - 50x² - 40x

Simplifying the expression by combining like terms, we get:

(fg)(x) = -50x² + 32

Step 2: Evaluate (fg)(-2)

Now that we have the expression for (fg)(x), we can evaluate it at x = -2. This means we substitute -2 for x in the expression:

(fg)(-2) = -50(-2)² + 32

Remember that (-2)² = (-2) * (-2) = 4, so we have:

(fg)(-2) = -50(4) + 32

(fg)(-2) = -200 + 32

(fg)(-2) = -168

Therefore, the value of (fg)(-2) is -168.

Alternative Method: Evaluating f(-2) and g(-2) First

There's also an alternative method to solve this problem. We can first evaluate f(-2) and g(-2) separately and then multiply the results.

Step 1: Evaluate f(-2)

Substitute x = -2 into the expression for f(x):

f(-2) = 8 - 10(-2)

f(-2) = 8 + 20

f(-2) = 28

Step 2: Evaluate g(-2)

Substitute x = -2 into the expression for g(x):

g(-2) = 5(-2) + 4

g(-2) = -10 + 4

g(-2) = -6

Step 3: Multiply f(-2) and g(-2)

Now, multiply the values we found for f(-2) and g(-2):

(fg)(-2) = f(-2) * g(-2) = 28 * (-6)

(fg)(-2) = -168

As you can see, this method also gives us the same answer: (fg)(-2) = -168.

Key Takeaways

• Function Composition: (fg)(x) represents the product of the functions f(x) and g(x).
• Function Evaluation: To evaluate a function at a specific value, substitute that value for the variable x in the function's expression.
• Order of Operations: Remember to follow the order of operations (PEMDAS/BODMAS) when evaluating expressions.
• Alternative Methods: There can be multiple ways to solve a problem. Understanding different approaches can enhance your problem-solving skills.

Practice Problems

To solidify your understanding, try solving these practice problems:

1. If f(x) = 3x² - 2 and g(x) = x + 5, find (fg)(1).
2. If f(x) = 4 - x and g(x) = 2x - 1, find (fg)(3).
3. If f(x) = x² + 1 and g(x) = -x + 2, find (fg)(-1).

By working through these problems, you'll gain confidence in your ability to solve function composition and evaluation problems.

Conclusion

In this article, we successfully determined the value of (fg)(-2) given the functions f(x) = 8 - 10x and g(x) = 5x + 4. We explored the concepts of function composition and evaluation, demonstrating the step-by-step process of finding (fg)(x) and then evaluating it at x = -2. We also presented an alternative method of solving the problem by evaluating f(-2) and g(-2) separately and then multiplying the results. By understanding these concepts and practicing regularly, you'll be well-equipped to tackle similar problems in mathematics. Remember that mastering functions is crucial for higher-level math, such as calculus, so building a strong foundation now will pay off in the long run. Function manipulation and evaluation are core skills, and problems like this help you strengthen those skills. Keep practicing, and you'll become more proficient in working with functions!

This exercise highlights the importance of understanding function operations and algebraic manipulation. By carefully applying the distributive property and combining like terms, we can accurately determine the value of combined functions at specific points. Function composition is a powerful tool in mathematics, allowing us to build complex models from simpler components. This concept is widely used in various fields, including physics, engineering, and computer science. The ability to work with functions is not just a mathematical skill; it's a valuable asset in problem-solving across different disciplines. The more you practice with functions, the more comfortable and confident you'll become in your mathematical abilities. Don't hesitate to explore different types of functions and their properties, as this will broaden your mathematical horizons and open doors to new possibilities.

Remember, mathematics is a journey of discovery, and every problem you solve is a step forward. Embrace the challenges, learn from your mistakes, and celebrate your successes. With dedication and perseverance, you can achieve your mathematical goals and unlock the beauty and power of this fascinating subject. Keep exploring, keep learning, and keep growing!