True Or False Is F(g(6)) Equal To 7

Introduction

In the realm of mathematics, composite functions play a crucial role in understanding the behavior of mathematical expressions. Composite functions, denoted as f(g(x)), involve the application of one function to the result of another. This operation can sometimes seem intricate, but it is essential for various mathematical applications. In this article, we delve into a specific problem involving composite functions to determine the truthfulness of a given statement. We will explore the functions f(x) = x² and g(x) = √{x + 5}, and evaluate whether f(g(6)) indeed equals 7. To achieve this, we will dissect the concept of composite functions, perform the necessary calculations, and provide a step-by-step explanation to ensure clarity and understanding. Whether you are a student grappling with this concept or a math enthusiast seeking to reinforce your knowledge, this exploration will offer valuable insights into the world of composite functions.

Understanding Composite Functions

To begin, let's clarify what a composite function is. A composite function is created when one function is plugged into another. Specifically, if we have two functions, f(x) and g(x), the composite function f(g(x)) means we first evaluate g(x) and then use the result as the input for f(x). This concept is fundamental in calculus, analysis, and many other branches of mathematics. It helps us understand how different mathematical operations can be combined to create more complex functions. To truly grasp composite functions, one must understand the order of operations and how the output of one function becomes the input of another. The domain and range of composite functions can be influenced by both the inner and outer functions, making it essential to consider both functions when analyzing composite functions. The composite function f(g(x)) is not always the same as g(f(x)), which demonstrates the importance of order in these operations. Mastering composite functions opens doors to understanding more complex mathematical models and transformations.

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

Before we tackle the specific problem, let's clearly define the functions we'll be working with. The first function is f(x) = x². This is a simple quadratic function, which squares whatever input it receives. For example, if we input 2 into f(x), we get f(2) = 2² = 4. The second function is g(x) = √{x + 5}. This is a square root function, where we first add 5 to the input and then take the square root. For instance, if we input 4 into g(x), we get g(4) = √{4 + 5} = √{9} = 3. Understanding these individual functions is crucial because the composite function will combine their operations. The function f(x) will take the output of g(x) as its input, thus linking the two functions. When working with radical functions like g(x), it is essential to be mindful of the domain, ensuring that the expression under the square root is non-negative. Both functions, f(x) and g(x), are essential building blocks in various mathematical contexts, and their combination in composite functions allows for richer mathematical expressions.

Evaluating g(6)

Now, let's begin our evaluation of the composite function. The first step in finding f(g(6)) is to determine the value of g(6). Recall that g(x) = √{x + 5}. To find g(6), we substitute 6 for x in the expression for g(x). This gives us g(6) = √{6 + 5}. Simplifying the expression inside the square root, we have g(6) = √{11}. The square root of 11 is an irrational number, and it is approximately 3.3166, but for the purpose of this evaluation, we will keep it in its exact form, √{11}. This value, √{11}, will now serve as the input for the function f(x). Understanding this step is crucial because it demonstrates how the output of the inner function becomes the input for the outer function in a composite function. This sequential evaluation is a hallmark of composite functions and requires careful attention to detail to avoid errors.

Evaluating f(g(6))

Having found that g(6) = √{11}, we now proceed to evaluate f(g(6)). Since g(6) = √{11}, we need to find f(√{11}). Recall that f(x) = x². To find f(√{11}), we substitute √{11} for x in the expression for f(x). This gives us f(√{11}) = (√{11})². When we square a square root, we simply get the number inside the square root, so (√{11})² = 11. Therefore, f(g(6)) = 11. This result demonstrates how the squaring operation in f(x) cancels out the square root operation (partially) in g(x). Understanding this interaction is essential for simplifying and evaluating composite functions effectively. The final value, 11, will be compared with the stated value of 7 to determine the truthfulness of the original statement.

Comparing the Result

Now that we have evaluated f(g(6)) and found it to be 11, we can compare this result to the statement given, which asserts that f(g(6)) = 7. Clearly, 11 is not equal to 7. This discrepancy is significant and indicates that the statement is false. The process of finding f(g(6)) involved two key steps: first, evaluating g(6), and then using that result as the input for f(x). Each step was carefully executed to arrive at the correct value of 11. The initial statement might have been a result of a calculation error or a misunderstanding of how composite functions operate. This comparison highlights the importance of verifying mathematical statements through step-by-step evaluation and logical reasoning. By identifying the discrepancy, we can confidently conclude that the original statement is false.

Conclusion: True or False?

In conclusion, we set out to determine whether the statement "If f(x) = x² and g(x) = √{x + 5}, then f(g(6)) = 7" is true or false. Through a step-by-step evaluation, we found that g(6) = √{11} and subsequently, f(g(6)) = f(√{11}) = 11. Comparing this result with the stated value of 7, we clearly see that f(g(6)) is not equal to 7. Therefore, the statement is false. Understanding how to evaluate composite functions is crucial in mathematics, and this exercise provides a clear example of the process. The key takeaway is that the order of operations matters, and each function must be applied correctly to arrive at the correct result. This exploration reinforces the importance of careful calculation and logical reasoning in mathematical problem-solving. Mastering the concept of composite functions not only helps in solving specific problems but also enhances the overall understanding of mathematical relationships and operations.