Decomposing |2(x+1)²+(x+1)| Identifying The Correct Function Pair

In the realm of mathematics, particularly in function analysis, the concept of function decomposition is crucial. It allows us to break down complex functions into simpler, more manageable components. This not only simplifies the analysis but also provides a deeper understanding of the function's behavior. In this article, we will delve into the process of decomposing a composite function, using the example f(g(x)) = |2(x+1)²+(x+1)|. We will explore various possible decompositions and rigorously determine the correct pair of functions that satisfy the given composite function. Our focus will be on understanding the underlying principles of function composition and how to effectively identify the inner and outer functions in a given composite function. This exploration will involve algebraic manipulation, pattern recognition, and careful consideration of the domain and range of the involved functions. Ultimately, we aim to provide a comprehensive guide to function decomposition, empowering readers to confidently tackle similar problems and enhance their mathematical problem-solving skills.

Function decomposition, also known as function composition, is the process of breaking down a complex function into simpler functions that, when composed together, yield the original function. A composite function is a function that is formed by applying one function to the result of another. In other words, if we have two functions, f(x) and g(x), the composite function f(g(x)) represents the function f applied to the result of g(x). The function g(x) is often referred to as the inner function, while f(x) is the outer function. The key to successful function decomposition lies in identifying these inner and outer functions within the given composite function. This often involves recognizing patterns, algebraic structures, and the order of operations. For instance, in the composite function f(g(x)), we first evaluate g(x) and then use that result as the input for f(x). Understanding this order is crucial for correctly decomposing the function. The ability to decompose functions is not only essential for simplifying mathematical expressions but also for solving complex problems in various fields, including calculus, differential equations, and computer science. By breaking down complex problems into smaller, more manageable parts, we can gain a deeper understanding of the underlying processes and develop effective solutions. Moreover, function decomposition plays a vital role in understanding the transformations of functions, such as shifts, stretches, and reflections, which are fundamental concepts in function analysis.

Our primary objective is to identify the pair of functions, f(x) and g(x), that accurately represent a decomposition of the composite function f(g(x)) = |2(x+1)²+(x+1)|. This means we need to find functions f(x) and g(x) such that when we substitute g(x) into f(x), the result is the given expression. The problem presents us with four potential pairs of functions, and our task is to systematically evaluate each pair to determine which one satisfies the composition condition. This process will involve substituting the expression for g(x) into f(x) and simplifying the resulting expression. We will then compare the simplified expression with the original composite function to see if they match. The correct pair of functions will be the one that, when composed, yields the exact expression |2(x+1)²+(x+1)|. This problem highlights the importance of understanding the order of operations in function composition and the ability to manipulate algebraic expressions. It also emphasizes the need for careful attention to detail and a systematic approach to problem-solving. By working through this problem, we will gain a deeper appreciation for the intricacies of function decomposition and its role in simplifying complex mathematical expressions. Furthermore, we will develop valuable skills in algebraic manipulation and function analysis that can be applied to a wide range of mathematical problems. The challenge lies in not only finding a pair that works but also in understanding why the other pairs do not, thus solidifying our grasp of the underlying concepts.

We are presented with four options, each proposing a pair of functions, f(x) and g(x), as potential decompositions of f(g(x)) = |2(x+1)²+(x+1)|. To determine the correct decomposition, we must meticulously examine each option, substituting g(x) into f(x) and simplifying the result. This process will reveal whether the composition of the proposed functions matches the given composite function. Let's break down each option and analyze it step by step:

Option A: f(x) = (x+1)² and g(x) = |2x+1|

In this case, we would substitute g(x) = |2x+1| into f(x) = (x+1)². This gives us f(g(x)) = (|2x+1|+1)². Expanding this expression, we get (|2x+1|+1)² = (2x+1)² + 2|2x+1| + 1 = 4x² + 4x + 1 + 2|2x+1| + 1 = 4x² + 4x + 2 + 2|2x+1|. This result does not immediately match the form of the original composite function, |2(x+1)²+(x+1)|, which suggests that this pair of functions is likely not the correct decomposition. However, we must consider that the absolute value within the original function might lead to different behaviors depending on the values of x. A more thorough comparison would involve analyzing specific cases or graphing the expressions to confirm the discrepancy.

Option B: f(x) = (x+1) and g(x) = |2x²+x|

Here, we substitute g(x) = |2x²+x| into f(x) = (x+1), resulting in f(g(x)) = |2x²+x| + 1. This expression clearly differs from the original composite function |2(x+1)²+(x+1)|. The additional '+ 1' term and the different algebraic structure indicate that this pair of functions does not represent the correct decomposition. The simplicity of this composition makes it relatively easy to rule out this option, as the resulting expression lacks the key components of the original function, such as the squared term within the absolute value.

Option C: f(x) = |2x+1| and g(x) = (x+1)²

In this option, we substitute g(x) = (x+1)² into f(x) = |2x+1|, giving us f(g(x)) = |2(x+1)²+1|. This expression is close to the original composite function |2(x+1)²+(x+1)|, but there's a crucial difference: the '+1' inside the absolute value instead of '+(x+1)'. This difference, though seemingly small, significantly alters the behavior of the function, particularly for certain values of x. Therefore, this pair of functions, while similar in structure, does not accurately represent the decomposition of the given composite function. The proximity of this option to the correct answer highlights the importance of careful observation and precise algebraic manipulation in function decomposition problems.

Option D: f(x) = |2x²+x| and g(x) = (x+1)

Finally, we substitute g(x) = (x+1) into f(x) = |2x²+x|. This yields f(g(x)) = |2(x+1)²+(x+1)|. This expression perfectly matches the original composite function. Therefore, this pair of functions represents the correct decomposition. The algebraic structure is identical, and the composition directly leads to the given expression. This confirms that option D is the solution to the problem.

To definitively determine the correct decomposition of the composite function f(g(x)) = |2(x+1)²+(x+1)|, we must meticulously evaluate each option by substituting g(x) into f(x) and comparing the result with the original expression. As we discussed in the analysis of the options, let's focus on the correct answer, option D, and demonstrate why it is the solution.

Option D: f(x) = |2x²+x| and g(x) = (x+1)

We begin by substituting g(x) = (x+1) into f(x) = |2x²+x|. This means we replace every instance of 'x' in f(x) with the expression for g(x), which is (x+1). This gives us:

f(g(x)) = |2(x+1)² + (x+1)|

This expression is precisely the original composite function that we were given. This direct match confirms that the functions f(x) = |2x²+x| and g(x) = (x+1) indeed represent a valid decomposition of f(g(x)) = |2(x+1)²+(x+1)|.

To further solidify our understanding, let's consider why the other options are incorrect. The key is to recognize the algebraic structure of the original function and how it arises from the composition of f(x) and g(x). The expression |2(x+1)²+(x+1)| contains a quadratic term, 2(x+1)², and a linear term, (x+1), both within the absolute value. This suggests that the inner function, g(x), should produce the '(x+1)' term, and the outer function, f(x), should handle the quadratic term and the absolute value. Options A, B, and C fail to achieve this specific structure in their composition.

In conclusion, the correct pair of functions that represents a decomposition of f(g(x)) = |2(x+1)²+(x+1)| is f(x) = |2x²+x| and g(x) = (x+1). This solution was obtained through a systematic process of substituting the proposed inner function g(x) into the outer function f(x) for each option and comparing the resulting composite function with the original expression. The ability to decompose composite functions is a fundamental skill in mathematics, requiring a solid understanding of function composition and algebraic manipulation. This problem highlights the importance of careful observation, precise calculations, and a systematic approach to problem-solving. By mastering these skills, one can confidently tackle more complex problems involving functions and their compositions. Furthermore, the process of analyzing incorrect options is equally important, as it reinforces the understanding of why certain compositions do not yield the desired result. This deeper understanding allows for a more intuitive approach to function decomposition, making it easier to identify the correct inner and outer functions in a given composite function. Ultimately, the ability to decompose functions is a valuable tool for simplifying complex mathematical expressions and gaining a deeper insight into the behavior of functions.