Composite Functions Explained Finding (g ⋅ F)(x) For F(x) = Log(5x) And G(x) = 5x + 4

In the realm of mathematics, particularly in the study of functions, composite functions play a crucial role in understanding how different functions interact and create new functions. A composite function is essentially a function that is applied to the result of another function. This means that the output of one function becomes the input of another. Understanding composite functions is essential for various mathematical concepts and applications, including calculus, algebra, and real-world modeling. This article will delve into the concept of composite functions, providing a detailed explanation and focusing on the specific example given: finding $(g \\cdot f)(x)$ when $f(x) = log(5x)$ and $g(x) = 5x + 4$. We will explore the process step-by-step, ensuring a clear understanding of how to evaluate and interpret composite functions. The notation $(g \\cdot f)(x)$ represents the composite function where the function $f$ is applied first, and then the function $g$ is applied to the result. This is a fundamental operation in mathematics, and mastering it opens the door to more advanced topics and problem-solving techniques. By the end of this discussion, you will not only be able to solve this particular problem but also gain a broader understanding of composite functions and their applications.

Defining Composite Functions

To truly grasp the concept of evaluating $(g \\cdot f)(x)$, it's essential to have a solid understanding of what composite functions are and how they operate. A composite function is created when one function is substituted into another. In mathematical terms, if we have two functions, $f(x)$ and $g(x)$, the composite function $(g \\cdot f)(x)$ is defined as $g(f(x))$. This means that we first apply the function $f$ to $x$, obtaining $f(x)$, and then we apply the function $g$ to the result $f(x)$. The order in which the functions are applied is crucial; $(g \\cdot f)(x)$ is generally different from $(f \\cdot g)(x)$, which is defined as $f(g(x))$. To visualize this, imagine a function machine. You input a value $x$ into the first machine, $f$, which transforms it into $f(x)$. Then, you take the output $f(x)$ and feed it into the second machine, $g$, which transforms it into $g(f(x))$. This two-step process is the essence of a composite function. Understanding this sequential application is key to correctly evaluating and interpreting composite functions. For example, consider the functions $f(x) = x^2$ and $g(x) = x + 1$. To find $(g \\cdot f)(x)$, we would first evaluate $f(x)$, which is $x^2$, and then substitute this result into $g(x)$, giving us $g(f(x)) = g(x^2) = x^2 + 1$. On the other hand, to find $(f \\cdot g)(x)$, we would first evaluate $g(x)$, which is $x + 1$, and then substitute this into $f(x)$, giving us $f(g(x)) = f(x + 1) = (x + 1)^2 = x^2 + 2x + 1$. As you can see, the order of the functions matters significantly. This foundational understanding is critical as we move forward to tackle the specific problem at hand.

Applying the Concept to the Given Functions

Now that we have a clear understanding of composite functions, let's apply this knowledge to the specific functions provided in the problem. We are given $f(x) = log(5x)$ and $g(x) = 5x + 4$, and our goal is to find the composite function $(g \\cdot f)(x)$. Remember, $(g \\cdot f)(x)$ means we need to substitute $f(x)$ into $g(x)$. In other words, we need to find $g(f(x))$. The first step is to identify what $f(x)$ is, which is given as $log(5x)$. This logarithmic function takes the input $x$, multiplies it by 5, and then calculates the logarithm of the result. The next step is to take this expression, $log(5x)$, and substitute it into the function $g(x)$. The function $g(x)$ is defined as $5x + 4$. This means that wherever we see an $x$ in the expression for $g(x)$, we need to replace it with $f(x)$, which is $log(5x)$. So, we have $g(f(x)) = 5(f(x)) + 4$. Now, we substitute $log(5x)$ for $f(x)$: $g(f(x)) = 5(log(5x)) + 4$. This is the composite function $(g \\cdot f)(x)$. It represents the function that results from first applying $f$ to $x$ and then applying $g$ to the result. It's crucial to follow this substitution process carefully to avoid errors. Understanding this step-by-step approach is essential for solving composite function problems effectively. We have now successfully constructed the composite function $(g \\cdot f)(x)$ using the given functions $f(x)$ and $g(x)$.

Evaluating the Composite Function (g ⋅ f)(x)

Having established the composite function $(g \\cdot f)(x) = 5(log(5x)) + 4$, the next step is to examine the result and ensure it aligns with the options provided. The expression $5(log(5x)) + 4$ represents the final form of the composite function. This function takes an input $x$, calculates the logarithm of $5x$, multiplies the result by 5, and then adds 4. It's important to note that this function is defined only for values of $x$ where $5x > 0$, since the logarithm of a non-positive number is undefined. This means that $x$ must be greater than 0 for the function to be valid. The expression $5(log(5x)) + 4$ is a single term, representing the entire composite function. There are no further simplifications possible without additional context or information. This is a crucial point to understand, as it helps in correctly identifying the answer from a set of options. Now, let's compare our result with the potential answers given in the original problem:

A. $(g \\cdot f)(x) = 5x \\log(5x) + 4$ B. $(g \\cdot f)(x) = 5x \\log(5x) + 4 \\log(5x)$ C. $(g \\cdot f)(x) = 5 \\log(5x) + 4$

By comparing our derived expression, $5(log(5x)) + 4$, with the options, we can see that option C matches our result exactly. The other options include terms like $5x \\log(5x)$ and $4 \\log(5x)$, which are not present in our derived expression. Therefore, the correct answer is C. This process of comparing the derived result with the given options is an essential step in problem-solving. It helps ensure that the final answer is accurate and aligns with the specific requirements of the problem.

Conclusion

In this detailed exploration, we have successfully navigated the process of finding the composite function $(g \\cdot f)(x)$ for the given functions $f(x) = log(5x)$ and $g(x) = 5x + 4$. We began by establishing a solid understanding of composite functions, emphasizing the importance of function order and the substitution process. We then applied this understanding to the specific problem, carefully substituting $f(x)$ into $g(x)$ to obtain $g(f(x)) = 5(log(5x)) + 4$. Finally, we compared our result with the provided options and correctly identified option C as the solution. This exercise highlights the importance of a systematic approach to solving mathematical problems. By breaking down the problem into smaller, manageable steps, we can tackle even complex problems with confidence. Understanding composite functions is a fundamental skill in mathematics, and this detailed explanation provides a strong foundation for further exploration of related concepts. The ability to evaluate and interpret composite functions is essential for various applications, including calculus, differential equations, and real-world modeling. By mastering this skill, you are well-equipped to tackle a wide range of mathematical challenges. Remember, the key to success in mathematics is not just memorizing formulas but also understanding the underlying concepts and applying them logically and systematically. This article has aimed to provide you with both the conceptual understanding and the practical skills needed to confidently work with composite functions.

Practice Problems

To solidify your understanding of composite functions, it's crucial to practice with various examples. Here are a few practice problems to help you further develop your skills:

1. Given $f(x) = x^2 + 1$ and $g(x) = 2x - 3$, find $(g \\cdot f)(x)$ and $(f \\cdot g)(x)$.
2. Given $f(x) = \\sqrt{x}$ and $g(x) = x + 5$, find $(g \\cdot f)(x)$ and $(f \\cdot g)(x)$.
3. Given $f(x) = \\frac{1}{x}$ and $g(x) = x^2 - 1$, find $(g \\cdot f)(x)$ and $(f \\cdot g)(x)$.

Working through these problems will help you internalize the process of evaluating composite functions and understand how the order of functions affects the final result. Remember to carefully substitute the functions and simplify the expressions. If you encounter any difficulties, revisit the explanations and examples provided in this article. With consistent practice, you will become proficient in working with composite functions and applying this knowledge to more complex mathematical problems.