Multiplying Functions Finding (f * G)(x) And (f * G)(-3)

Hey there, math enthusiasts! Today, we're going to embark on an exciting journey into the world of functions. Specifically, we'll be exploring the concept of the product of functions. We'll take two functions, f(x) and g(x), and learn how to find their product, denoted as (f * g)(x). But that's not all! We'll also dive into evaluating this product at a specific point, namely (f * g)(-3). So, buckle up and get ready to unravel the mysteries of function multiplication!

Defining Our Functions: f(x) and g(x)

Before we can delve into the product of functions, let's clearly define the functions we'll be working with. We are given two functions:

• f(x) = 3x² - 5x + 7
• g(x) = 3x + 9

f(x) is a quadratic function, characterized by the x² term, while g(x) is a linear function, as the highest power of x is 1. Understanding the nature of these functions is crucial as we proceed to find their product. The quadratic function f(x) will create a parabolic curve when graphed, influencing the overall shape of the resultant function (f * g)(x). The linear function g(x) will behave as a straight line when graphed and will add another dimension of complexity in the final function after multiplication. When we perform the multiplication, we must remember to distribute each term of the first function to every term of the second function, ensuring no terms are missed. This careful expansion is the cornerstone of determining the correct product. Moreover, paying close attention to the signs of the coefficients will help prevent the most common errors in these types of calculations. It is beneficial to rewrite the expression several times during the expansion process to confirm the correct distribution. By understanding the individual natures of f(x) and g(x), we lay the foundation for understanding the nature of their product, (f * g)(x).

Unveiling the Product: (f * g)(x)

The product of two functions, (f * g)(x), is simply the function obtained by multiplying the expressions for f(x) and g(x). So, in our case:

(f * g)(x) = f(x) * g(x) = (3x² - 5x + 7) * (3x + 9)

Now, we need to expand this product. This involves multiplying each term in the first expression by each term in the second expression. Let's break it down step-by-step:

1. Multiply 3x² by both terms in (3x + 9): 3x² * 3x = 9x³ and 3x² * 9 = 27x²
2. Multiply -5x by both terms in (3x + 9): -5x * 3x = -15x² and -5x * 9 = -45x
3. Multiply 7 by both terms in (3x + 9): 7 * 3x = 21x and 7 * 9 = 63

Now, let's put it all together:

(f * g)(x) = 9x³ + 27x² - 15x² - 45x + 21x + 63

Finally, we combine like terms to simplify the expression:

(f * g)(x) = 9x³ + (27x² - 15x²) + (-45x + 21x) + 63

(f * g)(x) = 9x³ + 12x² - 24x + 63

And there you have it! The product of f(x) and g(x), (f * g)(x), is the cubic function 9x³ + 12x² - 24x + 63. This cubic function (f * g)(x) exhibits characteristics that are influenced by both the original functions, quadratic f(x) and linear g(x). The presence of the x³ term indicates a more complex curve than either f(x) or g(x) individually would produce when graphed. Each term contributes to the overall behavior of the function, dictating its turns, intercepts, and asymptotic behavior. The calculation involved careful distribution and combination of like terms, a process vital in polynomial manipulation. Each step ensures that the resultant expression accurately represents the product of the two initial functions. In practical scenarios, understanding the product of functions can model real-world phenomena, where one aspect is described by f(x) and another by g(x), and their combined effect is captured by (f * g)(x). Such applications could range from physics to economics, making the grasp of this concept profoundly versatile.

Evaluating at x = -3: (f * g)(-3)

Now that we've found the general expression for (f * g)(x), let's evaluate it at a specific point, x = -3. This means we substitute -3 for every x in the expression we just found:

(f * g)(-3) = 9(-3)³ + 12(-3)² - 24(-3) + 63

Let's break down the calculation:

1. (-3)³ = -27
2. (-3)² = 9
3. 9 * (-27) = -243
4. 12 * 9 = 108
5. -24 * (-3) = 72

Now, substitute these values back into the expression:

(f * g)(-3) = -243 + 108 + 72 + 63

Finally, add the numbers together:

(f * g)(-3) = 0

So, the value of (f * g)(x) at x = -3 is 0. This implies that x = -3 is a root, or zero, of the function (f * g)(x). Geometrically, this corresponds to the point where the graph of (f * g)(x) crosses the x-axis. The process of evaluating a function at a specific point is fundamental in mathematics, science, and engineering. It allows us to predict the function's output for a given input, providing valuable insights into its behavior. In this case, finding (f * g)(-3) informs us about the function's nature near x = -3, which is valuable for graphing and analysis. The step-by-step calculation illustrates the meticulous process of substitution and arithmetic required to achieve the correct evaluation. By methodically working through each component, from the exponentiation to the final summation, we ensure the precision of the result. This skill is essential not only in theoretical mathematics but also in practical applications, where functions are used to model real-world scenarios and their evaluations provide critical data points.

Alternative Approach: Evaluating f(-3) and g(-3) First

There's another way to find (f * g)(-3), and it's often a good practice to know alternative approaches. Instead of finding (f * g)(x) first and then substituting x = -3, we can evaluate f(-3) and g(-3) separately and then multiply the results.

Let's find f(-3):

f(-3) = 3(-3)² - 5(-3) + 7 = 3(9) + 15 + 7 = 27 + 15 + 7 = 49

Now, let's find g(-3):

g(-3) = 3(-3) + 9 = -9 + 9 = 0

Finally, let's multiply the results:

(f * g)(-3) = f(-3) * g(-3) = 49 * 0 = 0

As you can see, we arrive at the same answer, 0, which confirms our previous calculation. This alternative method highlights a fundamental property of function multiplication: evaluating the product at a point is equivalent to multiplying the individual function values at that point. This property provides a powerful tool for checking results and can sometimes simplify calculations, especially when dealing with more complex functions. Evaluating f(-3) and g(-3) individually helps break down the problem into smaller, more manageable steps. The arithmetic involved in each evaluation, from squaring -3 to summing the terms, requires careful attention to detail. The ultimate multiplication of f(-3) and g(-3) underscores the core concept of function products. Choosing the most efficient method, whether it's finding (f * g)(x) first or evaluating individually, often depends on the specific functions involved and the context of the problem. Both methods offer valuable insights into the behavior of functions and their interactions, enriching our understanding of mathematical concepts.

In Conclusion: Mastering Function Multiplication

We've successfully navigated the realm of function multiplication! We started with two functions, f(x) and g(x), found their product (f * g)(x), and then evaluated this product at x = -3. We even explored an alternative method for evaluating (f * g)(-3). Understanding how to multiply functions is a crucial skill in mathematics, opening doors to more advanced concepts and applications. This journey through function multiplication underscores the importance of meticulous calculations, alternative problem-solving techniques, and the profound interconnectedness of mathematical ideas. The ability to confidently find and evaluate the product of functions empowers us to analyze complex systems, model real-world phenomena, and unlock deeper mathematical insights. As you continue your mathematical explorations, remember the versatility of functions and their ability to interact in various ways, enriching your problem-solving capabilities and broadening your mathematical perspective.

So, keep practicing, keep exploring, and keep having fun with math! You've got this! Remember to always double-check your work and explore different methods to solve problems. This not only helps you to catch any errors but also deepens your understanding of the concepts. Happy calculating!