Function Composition Find (g ∘ F)(a) Equals |a| - 2

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In the realm of mathematics, particularly in functional analysis, function composition is a fundamental operation that combines two functions to produce a new function. This article delves into the intricacies of function composition, focusing on the specific problem of identifying function pairs f(a) and g(a) that satisfy the condition (g ∘ f)(a) = |a| - 2. We will meticulously examine each option, providing step-by-step explanations and highlighting the key concepts involved. Understanding function composition is crucial not only for academic pursuits but also for various applications in computer science, engineering, and other scientific disciplines.

Understanding Function Composition

Before we dive into the specific problem, let's solidify our understanding of function composition. The composition of two functions, denoted as (g ∘ f)(a), is defined as g(f(a)). This means we first apply the function f to the input a, and then we apply the function g to the result. The order of operations is critical; (g ∘ f)(a) is generally not the same as (f ∘ g)(a). Function composition can be visualized as a chain of operations where the output of one function becomes the input of the next. This concept is widely used in computer programming where complex tasks are broken down into smaller, manageable functions that are composed to achieve the desired outcome. The domain and range of the resulting composite function are also important considerations. The domain of (g ∘ f)(a) is the set of all a in the domain of f such that f(a) is in the domain of g. The range of (g ∘ f)(a) is the set of all values g(f(a)) takes as a varies over the domain of (g ∘ f). Mastering function composition is essential for solving advanced mathematical problems and understanding various real-world phenomena modeled by mathematical functions. For instance, in physics, the trajectory of a projectile can be described by composing functions that represent the initial velocity, gravitational force, and air resistance. Similarly, in economics, supply and demand curves can be analyzed using function composition to determine market equilibrium. Thus, a thorough grasp of function composition is invaluable for students and professionals across diverse fields.

Problem Statement

Our main objective is to find the pair of functions f(a) and g(a) such that their composition, (g ∘ f)(a), equals |a| - 2, where |a| represents the absolute value of a. The absolute value function, |a|, is defined as a if a is non-negative and -a if a is negative. This function introduces a non-linearity that makes the problem interesting and requires careful consideration of different cases. The subtraction of 2 shifts the graph of |a| downwards by 2 units. Therefore, we are looking for functions f and g whose composition produces this transformed absolute value function. To solve this problem, we will evaluate (g ∘ f)(a) for each given pair of functions and compare the result with |a| - 2. This process involves substituting f(a) into g(a) and simplifying the expression. It's crucial to pay attention to the order of operations and any restrictions on the domains of the functions, such as square roots or denominators that cannot be zero. The problem highlights the interplay between different types of functions, including polynomials, square roots, and absolute values, and requires a strong understanding of their properties and transformations. Understanding the problem statement thoroughly is the first step towards finding the correct solution, as it sets the stage for the subsequent analytical steps. The challenge lies in identifying the specific functions that, when composed, yield the desired absolute value expression with a vertical shift. This requires a combination of algebraic manipulation, functional analysis, and attention to detail.

Analyzing Option A: f(a) = a² - 4 and g(a) = √a

Let's begin by analyzing option A, where f(a) = a² - 4 and g(a) = √a. To find (g ∘ f)(a), we substitute f(a) into g(a), resulting in g(f(a)) = √(f(a)) = √(a² - 4). Now, we need to determine if √(a² - 4) is equal to |a| - 2. The domain of g(a) = √a is a ≥ 0, so for g(f(a)) to be defined, we must have a² - 4 ≥ 0. This inequality holds when a ≤ -2 or a ≥ 2. Therefore, the domain of (g ∘ f)(a) in this case is (-∞, -2] ∪ [2, ∞). Now, let's consider the behavior of √(a² - 4) within its domain. When a is a large positive number, √(a² - 4) is approximately equal to a, which is consistent with |a| - 2 for large positive a. However, when a is a large negative number, √(a² - 4) is approximately equal to |a|, which is not equal to |a| - 2. This discrepancy suggests that option A is unlikely to be the correct answer. To further investigate, let's consider specific values of a. For example, if a = 3, then √(a² - 4) = √(3² - 4) = √5, while |a| - 2 = |3| - 2 = 1. Since √5 ≠ 1, we can confidently conclude that option A is not the correct pair of functions. This analysis highlights the importance of considering both the algebraic expression and the domain of the composite function. The presence of a square root introduces domain restrictions that must be carefully accounted for when determining the equality of functions. The algebraic manipulation and domain analysis in this step demonstrate the key skills required to solve function composition problems effectively.

Analyzing Option B: f(a) = (1/2)a - 1 and g(a) = 2a - 2

Next, we examine option B, where f(a) = (1/2)a - 1 and g(a) = 2a - 2. To find (g ∘ f)(a), we substitute f(a) into g(a), resulting in g(f(a)) = 2(f(a)) - 2 = 2((1/2)a - 1) - 2. Simplifying this expression, we get 2((1/2)a - 1) - 2 = a - 2 - 2 = a - 4. Now, we need to check if a - 4 is equal to |a| - 2. Clearly, a - 4 is a linear function, while |a| - 2 is an absolute value function. These two functions have different forms and behaviors. For instance, when a = 0, a - 4 = -4, but |a| - 2 = |0| - 2 = -2. Since -4 ≠ -2, we can definitively say that option B is not the correct answer. This analysis demonstrates the importance of recognizing the types of functions involved. Linear functions and absolute value functions have distinct properties and graphs. A linear function is a straight line, while the absolute value function has a V-shape. The composition of two linear functions will always result in another linear function, which cannot be equal to an absolute value function. This understanding helps us quickly eliminate options that are clearly not the solution. The simplicity of the functions in option B allows for a straightforward algebraic simplification and comparison, highlighting the efficiency of this approach when dealing with linear functions. The direct comparison of the resulting expression with the target function provides a clear and concise way to rule out this option.

Analyzing Option C: f(a) = 5 + a² and g(a) = √(a - 5) - 2

Now, let's analyze option C, where f(a) = 5 + a² and g(a) = √(a - 5) - 2. To find (g ∘ f)(a), we substitute f(a) into g(a), resulting in g(f(a)) = √(f(a) - 5) - 2 = √((5 + a²) - 5) - 2. Simplifying this expression, we get √((5 + a²) - 5) - 2 = √(a²) - 2. Since √(a²) = |a|, we have √(a²) - 2 = |a| - 2. This result matches the desired expression |a| - 2. Therefore, option C is the correct pair of functions. The domain of f(a) = 5 + a² is all real numbers, and the range is [5, ∞). The domain of g(a) = √(a - 5) - 2 is a ≥ 5, which is consistent with the range of f(a). This confirms that the composition (g ∘ f)(a) is well-defined for all real numbers. The key step in this analysis is recognizing that √(a²) = |a|. This identity is crucial for simplifying the expression and obtaining the desired absolute value function. The successful match in option C underscores the importance of carefully simplifying the composite function and recognizing fundamental mathematical identities. This option demonstrates how a square root function and a quadratic function can combine to produce an absolute value function, highlighting the richness and versatility of function composition. The analysis of option C provides a clear and concise solution to the problem, showcasing the effectiveness of the step-by-step approach.

Analyzing Option D: f(a) = 3 - 3a and g(a) = 4a - 5

Finally, we analyze option D, where f(a) = 3 - 3a and g(a) = 4a - 5. To find (g ∘ f)(a), we substitute f(a) into g(a), resulting in g(f(a)) = 4(f(a)) - 5 = 4(3 - 3a) - 5. Simplifying this expression, we get 4(3 - 3a) - 5 = 12 - 12a - 5 = 7 - 12a. Now, we need to check if 7 - 12a is equal to |a| - 2. Similar to option B, we have a linear function 7 - 12a and an absolute value function |a| - 2. These functions have different forms and behaviors. For example, when a = 0, 7 - 12a = 7, but |a| - 2 = |0| - 2 = -2. Since 7 ≠ -2, we can conclude that option D is not the correct answer. This analysis reinforces the understanding that the composition of linear functions will always result in another linear function, which cannot be equal to an absolute value function. The straightforward algebraic simplification and comparison in this case provide a quick and efficient way to eliminate option D. The difference in the functional forms between the linear result and the target absolute value function is a clear indicator that this option is not the solution. The consistent application of the function composition process across all options demonstrates the methodical approach required to solve such problems effectively.

In conclusion, after meticulously analyzing all the given options, we have determined that the pair of functions for which (g ∘ f)(a) = |a| - 2 is option C: f(a) = 5 + a² and g(a) = √(a - 5) - 2. This problem highlights the importance of understanding function composition, domain restrictions, and algebraic simplification. By systematically evaluating each option, we were able to identify the correct pair of functions that satisfy the given condition. Function composition is a fundamental concept in mathematics with wide-ranging applications, making it essential for students and professionals in various fields. The ability to manipulate and analyze composite functions is crucial for solving complex problems and modeling real-world phenomena. The process of elimination employed in this article demonstrates a powerful problem-solving strategy that can be applied to various mathematical contexts. The successful identification of the correct answer reinforces the significance of a thorough understanding of function properties and algebraic techniques. This detailed exploration provides a comprehensive guide to solving function composition problems, emphasizing the importance of careful analysis and methodical execution.