Composite Functions Step By Step Solutions For F(x) = √x And G(x) = X-6

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In the realm of mathematics, composite functions offer a fascinating way to combine two functions, creating a new function that represents the sequential application of the original functions. This article aims to provide a comprehensive guide to understanding composite functions, focusing on the notation, evaluation, and practical application. We will explore the concepts using a specific example, breaking down each step to ensure clarity. Let's consider the functions f(x) = √x and g(x) = x - 6, and we will delve into finding (f∘g)(x), (g∘f)(x), and (f∘g)(10).

What are Composite Functions?

Composite functions are essentially functions within functions. Imagine you have two machines: one that doubles a number and another that adds 3 to the result. A composite function would be like feeding a number into the first machine, and then feeding the output of that machine into the second machine. Mathematically, this is represented as (f∘g)(x) or (g∘f)(x), where the circle symbol (∘) denotes function composition. The order matters significantly, as (f∘g)(x) is generally not the same as (g∘f)(x). To truly grasp the essence of composite functions, it's crucial to understand that the output of one function becomes the input of the other. This sequential application is what defines the nature of function composition.

Part a: Finding (f∘g)(x)

Our primary goal here is to determine (f∘g)(x). Understanding (f∘g)(x) is key, and it means we first apply the function g to x, and then apply the function f to the result. This can be written as f(g(x)). So, let's break it down step-by-step:

  1. Identify g(x): We are given that g(x) = x - 6.
  2. Substitute g(x) into f(x): This means we replace the 'x' in f(x) with the entire expression for g(x). Since f(x) = √x, we have f(g(x)) = √(g(x)) = √(x - 6).

Therefore, (f∘g)(x) = √(x - 6). This resulting function represents the composite function where we first subtract 6 from x, and then take the square root of the result. The simplicity of this process lies in understanding the order of operations – applying g first, then f. This step-by-step approach is fundamental to mastering composite functions. In essence, we've created a new function that encapsulates the combined actions of both f and g. The expression √(x - 6) is the simplified form of the composite function (f∘g)(x), ready for further analysis or evaluation.

Part b: Finding (g∘f)(x)

Now, let's shift our focus to finding (g∘f)(x). This composite function (g∘f)(x) signifies that we first apply the function f to x, and then apply the function g to the result. This is mathematically expressed as g(f(x)). It's crucial to recognize the difference in order compared to (f∘g)(x), as the outcome will generally be different. Let's proceed with the steps:

  1. Identify f(x): We know that f(x) = √x.
  2. Substitute f(x) into g(x): We replace the 'x' in g(x) with the expression for f(x). Given g(x) = x - 6, we have g(f(x)) = f(x) - 6 = √x - 6.

Consequently, (g∘f)(x) = √x - 6. This composite function illustrates the process of first taking the square root of x, and then subtracting 6 from the result. The order of function application is paramount, and in this case, it leads to a different composite function than we obtained in part a. The resulting expression, √x - 6, is the simplified form of (g∘f)(x). This clearly demonstrates how changing the order of composition alters the resulting function. Understanding this principle is essential for working with and manipulating composite functions effectively.

Part c: Evaluating (f∘g)(10)

Our final task is to evaluate (f∘g)(10). Evaluating (f∘g)(10) involves substituting x = 10 into the composite function we found in part a, which was (f∘g)(x) = √(x - 6). This process demonstrates how we can use composite functions to perform calculations for specific input values. Let's follow the steps:

  1. Recall (f∘g)(x): We determined that (f∘g)(x) = √(x - 6).
  2. Substitute x = 10: Replace 'x' with 10 in the expression: (f∘g)(10) = √(10 - 6).
  3. Simplify: (f∘g)(10) = √4 = 2.

Therefore, (f∘g)(10) = 2. This result indicates the output of the composite function when the input is 10. We first subtracted 6 from 10, resulting in 4, and then took the square root of 4, which gave us 2. This evaluation process highlights the practical application of composite functions in determining outputs for given inputs. By substituting the specific value into the simplified composite function, we efficiently calculated the result. This is a fundamental skill in working with mathematical functions and their compositions.

Summarizing the Findings

In this exploration of composite functions, we have successfully determined the following:

  • (f∘g)(x) = √(x - 6): This represents the composite function where g(x) is applied first, followed by f(x).
  • (g∘f)(x) = √x - 6: This shows the composite function where f(x) is applied first, followed by g(x).
  • (f∘g)(10) = 2: This is the value of the composite function (f∘g)(x) when x is 10.

These results demonstrate the process of function composition and evaluation. The key takeaway is the importance of the order of operations when dealing with composite functions. The ability to find and evaluate composite functions is a valuable tool in various mathematical and scientific contexts.

Key Concepts and Applications of Composite Functions

Key Concepts of Composite Functions

Delving deeper into the key concepts of composite functions, we find that they are not just mathematical curiosities but fundamental tools with widespread applications. The core idea is that a composite function combines two functions sequentially. When we write (f∘g)(x), we are essentially saying,