Evaluating (f * G)(-2) Where F(x) = 10x And G(x) = 5x + 4

In the realm of mathematics, functions serve as fundamental building blocks, mapping inputs to corresponding outputs. When we encounter composite functions, denoted as (f * g)(x), we delve into the intricate interplay between these mathematical entities. This article aims to elucidate the process of evaluating composite functions, specifically focusing on the scenario where f(x) = 10x and g(x) = 5x + 4, and we seek to determine the value of (f * g)(-2). Let's embark on this mathematical journey, step by step, to unravel the solution.

Understanding Composite Functions

Before we delve into the specifics of the given problem, let's first establish a firm grasp of what composite functions entail. A composite function arises when one function acts upon the result of another. In essence, we are plugging one function into another. The notation (f * g)(x) signifies that we are first applying the function g to the input x, and then subsequently applying the function f to the result obtained from g(x). In mathematical terms, (f * g)(x) = f(g(x)).

To illustrate this concept, consider two functions, f(x) and g(x). When we form the composite function (f * g)(x), we initially evaluate g(x), which yields a certain value. This value then becomes the input for the function f, and we evaluate f at this value. The final result is the output of the composite function (f * g)(x). Understanding this sequential process is crucial for accurately evaluating composite functions.

The order in which we apply the functions is paramount. (f * g)(x) is generally not the same as (g * f)(x), highlighting the importance of adhering to the correct order of operations. Visualizing composite functions as a chain of operations can aid in comprehending their behavior. The input x initiates the chain, passes through g, and then flows through f, ultimately producing the output of the composite function. This chain-like analogy underscores the sequential nature of composite function evaluation.

Decoding f(x) = 10x and g(x) = 5x + 4

Now that we have a solid understanding of composite functions, let's turn our attention to the specific functions provided in the problem: f(x) = 10x and g(x) = 5x + 4. These are linear functions, characterized by their straight-line graphs. Understanding the behavior of these individual functions is essential for evaluating their composite function.

The function f(x) = 10x represents a simple scaling operation. It takes an input x and multiplies it by 10. This means that for any input value, the output will be ten times that value. For instance, if x = 2, then f(2) = 10 * 2 = 20. This function stretches the input value by a factor of 10, effectively amplifying it.

On the other hand, the function g(x) = 5x + 4 involves both scaling and translation. It first multiplies the input x by 5, and then adds 4 to the result. This means that for any input value, the output will be five times that value, shifted upwards by 4 units. For example, if x = 2, then g(2) = 5 * 2 + 4 = 14. This function stretches the input value by a factor of 5 and then moves it upwards by 4 units on the coordinate plane.

Individually, these functions perform relatively straightforward operations. However, when combined in a composite function, their interplay becomes more nuanced. The output of g(x) becomes the input for f(x), leading to a sequence of operations that ultimately determine the output of the composite function. Understanding the individual behaviors of f(x) and g(x) is crucial for predicting the behavior of their composite function.

Evaluating (f * g)(-2): A Step-by-Step Approach

With a firm grasp of composite functions and the individual functions f(x) and g(x), we are now well-equipped to tackle the core of the problem: evaluating (f * g)(-2). This entails finding the value of the composite function (f * g)(x) when x = -2. To accomplish this, we will follow a systematic, step-by-step approach, ensuring clarity and accuracy in our calculations.

Step 1: Evaluate g(-2)

The first step in evaluating (f * g)(-2) is to determine the value of g(-2). This means substituting -2 for x in the expression for g(x): g(-2) = 5(-2) + 4. Performing the arithmetic, we get g(-2) = -10 + 4 = -6. This result tells us that when the input to g(x) is -2, the output is -6. This value will now serve as the input for the function f.

Step 2: Evaluate f(g(-2))

The second step involves evaluating f(g(-2)). Since we have already determined that g(-2) = -6, we can rewrite this as f(-6). This means substituting -6 for x in the expression for f(x): f(-6) = 10(-6). Performing the multiplication, we obtain f(-6) = -60. This result signifies that when the input to f(x) is -6, the output is -60.

Step 3: Conclude the Result

Having evaluated both g(-2) and f(g(-2)), we can now confidently state the value of (f * g)(-2). Since (f * g)(-2) = f(g(-2)) and we found that f(g(-2)) = -60, we conclude that (f * g)(-2) = -60. This is the final answer to the problem. The composite function (f * g)(x), when evaluated at x = -2, yields the value -60.

Alternative Approach: Deriving the Composite Function First

While we successfully evaluated (f * g)(-2) by first finding g(-2) and then using that result as the input for f(x), there exists an alternative approach that involves deriving the composite function (f * g)(x) explicitly before substituting the value of x. This method can be particularly useful when we need to evaluate the composite function for multiple values of x, as it avoids repeating the function composition process for each value.

Step 1: Find the expression for (f * g)(x)

To find the expression for (f * g)(x), we need to substitute the expression for g(x) into the function f(x). Recall that f(x) = 10x and g(x) = 5x + 4. Therefore, (f * g)(x) = f(g(x)) = f(5x + 4). Now, we substitute (5x + 4) for x in the expression for f(x): f(5x + 4) = 10(5x + 4). Distributing the 10, we get (f * g)(x) = 50x + 40. This is the explicit expression for the composite function (f * g)(x).

Step 2: Evaluate (f * g)(-2)

Now that we have the expression for (f * g)(x), we can directly substitute x = -2 into the expression: (f * g)(-2) = 50(-2) + 40. Performing the arithmetic, we get (f * g)(-2) = -100 + 40 = -60. This result matches the answer we obtained using the previous method, reinforcing the correctness of our solution.

This alternative approach highlights the flexibility in evaluating composite functions. Depending on the specific problem and the context, either method – evaluating step-by-step or deriving the composite function first – can be employed to arrive at the solution. The key is to choose the method that best suits the given situation and to execute the steps with precision and clarity.

Conclusion: Mastering Composite Function Evaluation

In this comprehensive exploration, we have successfully unraveled the value of (f * g)(-2) when f(x) = 10x and g(x) = 5x + 4. We began by establishing a firm understanding of composite functions, emphasizing the sequential nature of their evaluation. We then delved into the specific functions provided, dissecting their individual behaviors. Through a step-by-step approach, we meticulously evaluated g(-2) and subsequently used that result to evaluate f(g(-2)), ultimately arriving at the solution (f * g)(-2) = -60.

Furthermore, we explored an alternative method that involved deriving the explicit expression for the composite function (f * g)(x) before substituting the value of x. This approach showcased the versatility in tackling composite function problems and reinforced the accuracy of our solution. By mastering these techniques, one can confidently navigate the realm of composite functions and adeptly evaluate them for various input values.

The significance of composite functions extends far beyond the realm of textbook exercises. They serve as powerful tools for modeling real-world phenomena, where one process depends on the outcome of another. From physics to economics, composite functions find applications in diverse fields, underscoring their importance in mathematical analysis. A thorough understanding of composite functions empowers individuals to analyze complex systems and make informed decisions based on mathematical models.

As we conclude this exploration, let us reiterate the key takeaways. Composite functions involve applying one function to the result of another, and the order of application is crucial. Evaluating composite functions can be achieved through a step-by-step approach or by deriving the explicit expression for the composite function. The choice of method depends on the specific problem and the desired level of generality. With a solid grasp of these concepts, one can confidently tackle composite function problems and appreciate their significance in mathematics and beyond.