Solving For B In Composite Functions A Mathematical Exploration

In the captivating realm of mathematics, functions serve as fundamental building blocks, mapping inputs to outputs and weaving intricate relationships between variables. This article delves into a fascinating problem involving composite functions, where the output of one function becomes the input of another. Our mission is to unravel the mystery and determine the value of a specific parameter, 'b,' that satisfies a given condition. Prepare to embark on a mathematical journey that will challenge your analytical skills and deepen your appreciation for the elegance of functional relationships.

At the heart of our puzzle lie two distinct functions, each with its unique personality and behavior. The first function, denoted as f(x), is defined by the equation f(x) = (17.8x^2 + b) / 3. This function takes an input value 'x,' squares it, multiplies it by 17.8, adds the constant 'b,' and finally divides the result by 3. The presence of the parameter 'b' introduces an element of flexibility, allowing us to adjust the function's output based on its value. Understanding how 'b' influences f(x) is crucial to solving our problem. Our investigation of the function f(x) reveals that it is a quadratic function, characterized by the x^2 term. This means that the graph of f(x) will be a parabola, a U-shaped curve that can open upwards or downwards depending on the coefficient of the x^2 term. In our case, the coefficient 17.8 is positive, indicating that the parabola opens upwards. The parameter 'b' plays a significant role in shifting the parabola vertically. A larger value of 'b' will shift the parabola upwards, while a smaller value will shift it downwards. This vertical shift directly affects the output of f(x) for any given input 'x'. The second function, g(x), is more enigmatic. We are only given a single piece of information about g(x): that g(b) = 10. This tells us that when the input to g(x) is 'b,' the output is 10. However, we do not have a complete definition of g(x), which adds an element of intrigue to our task. Without knowing the explicit form of g(x), we must rely on the given information and our understanding of composite functions to proceed. The fact that g(b) = 10 provides a crucial link between the two functions, as it establishes a specific input-output relationship for g(x) that we can leverage in our calculations.

Now, let's introduce the concept of a composite function. A composite function is formed when we apply one function to the result of another. In our case, we are interested in the composite function f(g(b)). This means that we first evaluate the function g(x) at the input 'b,' obtaining g(b), and then we use this result as the input for the function f(x). In essence, f(g(b)) is a chain reaction, where the output of g(x) becomes the input of f(x). The order of operations is critical in composite functions. We must first evaluate the inner function, g(x), and then use its output as the input for the outer function, f(x). This process of chaining functions together allows us to create complex relationships between variables and explore the interplay between different mathematical operations. The given information that g(b) = 10 is pivotal in evaluating the composite function f(g(b)). Since we know the output of g(x) when the input is 'b,' we can substitute this value into f(x) and simplify the expression. This substitution transforms the problem into a more manageable form, where we can focus on solving for the unknown parameter 'b'.

The heart of our problem lies in the condition f(g(b)) = 600. This equation tells us that the output of the composite function f(g(b)) must be equal to 600. This condition provides a crucial constraint that will allow us to determine the value of 'b.' By setting up this equation, we establish a mathematical relationship that 'b' must satisfy. Solving this equation will lead us to the solution of our problem. The equation f(g(b)) = 600 represents a target value for the composite function. We are seeking a value of 'b' that, when plugged into g(x) and then f(x), will produce an output of exactly 600. This target value acts as a guide, helping us navigate the functional relationships and identify the specific value of 'b' that meets the required condition. The fact that the target value is a relatively large number (600) suggests that the value of 'b' might also be significant. This intuition can help us in our problem-solving process, allowing us to make informed guesses and check our results.

Now, let's embark on the journey of solving for 'b.' We will employ a step-by-step approach, carefully substituting and simplifying expressions until we isolate the value of 'b.' Our first step is to substitute g(b) = 10 into the composite function f(g(b)). This gives us f(10) = 600. We have now simplified the problem to a single equation involving only the function f(x) and the unknown 'b.' Next, we substitute x = 10 into the definition of f(x): f(10) = (17.8 * 10^2 + b) / 3. This step replaces the function notation with the explicit algebraic expression, allowing us to perform calculations and manipulate the equation. Now we have the equation (17.8 * 10^2 + b) / 3 = 600. This equation is our key to unlocking the value of 'b.' To solve for 'b,' we will perform a series of algebraic manipulations. First, we multiply both sides of the equation by 3: 17.8 * 10^2 + b = 1800. This step eliminates the fraction and simplifies the equation further. Next, we calculate 17.8 * 10^2, which equals 1780. This gives us the equation 1780 + b = 1800. Finally, we subtract 1780 from both sides of the equation to isolate 'b': b = 1800 - 1780. This yields the solution b = 20. Therefore, the value of 'b' that satisfies the condition f(g(b)) = 600 is 20. We have successfully navigated the functional relationships and algebraic manipulations to arrive at our answer.

As a final step, it is crucial to verify our solution. To do this, we will substitute b = 20 back into the original equation and check if the condition f(g(b)) = 600 is satisfied. First, we calculate g(20), which is given as 10. Next, we calculate f(10) using the definition of f(x) with b = 20: f(10) = (17.8 * 10^2 + 20) / 3. This simplifies to f(10) = (1780 + 20) / 3 = 1800 / 3 = 600. Since f(g(20)) = f(10) = 600, our solution b = 20 is verified. This verification step is essential to ensure that we have not made any errors in our calculations and that our solution is indeed correct. By substituting the value of 'b' back into the original equation, we confirm that the condition f(g(b)) = 600 is satisfied, providing us with confidence in our answer. The process of verification reinforces our understanding of the problem and the solution, solidifying our mathematical reasoning.

In this mathematical exploration, we successfully determined the value of 'b' that satisfies the condition f(g(b)) = 600. By carefully analyzing the functions f(x) and g(x), understanding the concept of composite functions, and employing a step-by-step algebraic approach, we unraveled the puzzle and arrived at the solution b = 20. This problem exemplifies the beauty and power of mathematics in describing and solving intricate relationships between variables. The journey of solving this problem has highlighted the importance of functional relationships, composite functions, and algebraic manipulations. By mastering these concepts, we can tackle a wide range of mathematical challenges and deepen our appreciation for the elegance of mathematical reasoning. The successful resolution of this problem serves as a testament to our analytical skills and our ability to navigate the world of mathematical puzzles. As we continue our mathematical journey, we will encounter many more fascinating problems that will challenge our minds and expand our understanding of the world around us.