Finding Values Of A And B In Composite Functions - A Detailed Solution
Introduction
In the realm of mathematical functions, composite functions play a vital role in understanding the relationships between different expressions. This article delves into the intricacies of composite functions, specifically focusing on how to determine unknown coefficients within these functions. Our central problem involves two functions, f(x) and g(x), defined as f(x) = a/(3 - x) and g(x) = 11 - bx², where a and b are constants we aim to find. We are given two crucial pieces of information: f(4) = 4/3 and f(g(2)) = 1/2. By leveraging these conditions and the principles of composite functions, we will systematically unravel the values of a and b, providing a clear and comprehensive solution.
Understanding Composite Functions
Before diving into the solution, let's solidify our understanding of composite functions. A composite function, denoted as (f ◦ g)(x) or f(g(x)), represents the application of one function to the result of another. In simpler terms, we first evaluate the inner function, g(x), and then use its output as the input for the outer function, f(x). This process creates a chain of operations, linking the two functions together. Understanding this chain is key to solving problems involving composite functions. For instance, if we have f(x) = x + 1 and g(x) = x², then f(g(x)) = f(x²) = x² + 1. The order of operations is crucial; f(g(x)) is generally not the same as g(f(x)). This non-commutative property adds a layer of complexity and necessitates careful attention to the order in which the functions are applied. Composite functions are fundamental in various mathematical contexts, including calculus, where they are essential for understanding the chain rule of differentiation. They also appear frequently in real-world applications, such as modeling complex systems where multiple processes interact sequentially. Therefore, mastering the concept of composite functions is not only crucial for solving mathematical problems but also for applying mathematical principles to practical scenarios. This article aims to enhance your understanding of composite functions by walking you through a detailed solution to a specific problem, highlighting the key steps and reasoning involved in finding unknown coefficients within these functions. By the end of this article, you should have a solid grasp of how to approach similar problems and apply the concepts of composite functions effectively.
Step-by-Step Solution
1. Finding the Value of 'a'
Our first task is to determine the value of a using the given information f(4) = 4/3. We know that f(x) = a/(3 - x), so we can substitute x with 4:
f(4) = a/(3 - 4) = a/(-1) = -a
Since we are given that f(4) = 4/3, we can set up the equation:
-a = 4/3
Solving for a, we multiply both sides by -1:
a = -4/3
Therefore, the value of a is -4/3. This result is crucial as it allows us to define the function f(x) more precisely. Now we know that f(x) = (-4/3)/(3 - x). With the value of a determined, we can proceed to the next step, which involves finding the value of b. The information provided about the composite function f(g(2)) will be instrumental in this endeavor. By substituting the value of a into the function f(x) and then using the properties of composite functions, we can set up an equation that will allow us to solve for b. The process involves careful substitution and algebraic manipulation, ensuring that we follow the correct order of operations. This step-by-step approach is essential for tackling complex problems involving composite functions, as it breaks down the problem into manageable parts, making the solution more accessible and understandable. The determination of a is not just an isolated step; it is a foundational element that paves the way for the subsequent steps in solving the overall problem. Without knowing the value of a, we would be unable to proceed with finding b. Thus, this initial step highlights the importance of systematic problem-solving, where each step builds upon the previous one, leading to the final solution.
2. Finding the Value of 'b'
Now, let's find the value of b using the information f(g(2)) = 1/2. We know that g(x) = 11 - bx², so first, we need to find g(2):
g(2) = 11 - b(2)² = 11 - 4b
Next, we substitute g(2) into f(x). We already know that f(x) = (-4/3)/(3 - x), so:
f(g(2)) = f(11 - 4b) = (-4/3) / (3 - (11 - 4b))
Simplifying the denominator:
f(g(2)) = (-4/3) / (3 - 11 + 4b) = (-4/3) / (-8 + 4b)
We are given that f(g(2)) = 1/2, so we set up the equation:
( -4/3 ) / ( -8 + 4b ) = 1/2_
To solve for b, we can cross-multiply:
2 * ( -4/3 ) = 1 * ( -8 + 4b )
-8/3 = -8 + 4b
Add 8 to both sides:
-8/3 + 8 = 4b
-8/3 + 24/3 = 4b
16/3 = 4b
Divide both sides by 4:
b = (16/3) / 4 = 16 / (3 * 4) = 16 / 12 = 4/3
Therefore, the value of b is 4/3. This completes our solution, as we have now found both a and b. The process of finding b involved several steps, including evaluating g(2), substituting the result into f(x), simplifying the expression, and finally solving the resulting equation. Each step required careful attention to detail to avoid errors in algebraic manipulation. The use of cross-multiplication and the subsequent simplification of fractions are essential skills in solving equations of this type. The final value of b is crucial for understanding the behavior of the function g(x), and together with the value of a, it provides a complete picture of the two functions involved in the composite function. This step-by-step approach not only leads to the correct solution but also enhances the understanding of the underlying concepts and techniques involved in solving composite function problems.
Conclusion
In conclusion, we have successfully determined the values of a and b in the given functions. We found that a = -4/3 and b = 4/3. This process involved a clear, step-by-step approach, utilizing the given information and the principles of composite functions. Understanding how to solve such problems is crucial for mastering the concepts of functions and their compositions, which are fundamental in mathematics.
By systematically working through the problem, we first used the information about f(4) to find the value of a. This was a straightforward substitution and algebraic manipulation, but it laid the groundwork for the next step. Once we had the value of a, we could then use the information about f(g(2)) to find the value of b. This involved evaluating g(2), substituting the result into f(x), simplifying the expression, and solving the resulting equation. This process highlighted the importance of careful attention to detail and the correct order of operations. The values of a and b are not just numerical answers; they are essential parameters that define the behavior of the functions f(x) and g(x). Knowing these values allows us to analyze and predict the output of these functions for any given input. Furthermore, understanding how to find these parameters is a valuable skill in mathematics, as it demonstrates the ability to apply algebraic techniques to solve real-world problems. The problem-solving approach we used in this article can be applied to a wide range of similar problems involving composite functions. The key is to break down the problem into manageable steps, carefully substitute the given information, and use algebraic manipulation to solve for the unknown variables. This systematic approach not only leads to the correct solution but also enhances the understanding of the underlying mathematical concepts. In essence, this exercise demonstrates the power of combining the principles of composite functions with algebraic techniques to solve for unknown parameters, a fundamental skill in mathematics and its applications.
