Evaluating F(-1) For The Function F(x) = 6(5)^2 A Step-by-Step Guide

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In the fascinating world of mathematics, functions serve as fundamental building blocks, describing relationships between variables and providing a powerful framework for modeling real-world phenomena. Understanding how to evaluate functions is a crucial skill, allowing us to unravel their behavior and extract valuable insights. In this comprehensive article, we embark on a journey to delve into the intricacies of function evaluation, specifically focusing on the function f(x) = 6(5)^2. Our primary objective is to determine the value of f(-1), a task that will illuminate the core principles of function manipulation and substitution. This exploration will not only enhance your mathematical prowess but also equip you with the tools to tackle a wide array of function-related challenges. We'll start by dissecting the given function, understanding its structure and the role of each component. Then, we'll proceed to the crucial step of substitution, replacing the variable x with the specified value, -1. Finally, we'll carefully perform the necessary arithmetic operations to arrive at the final, simplified exact value. Along the way, we'll emphasize the importance of order of operations and provide clear, step-by-step explanations to ensure a thorough understanding. Whether you're a student grappling with function concepts or a seasoned math enthusiast seeking a refresher, this article promises to be an enlightening and rewarding experience. Prepare to unlock the secrets of function evaluation and witness the elegance of mathematical expressions.

Before we can embark on the journey of evaluating f(-1), it's crucial to thoroughly understand the function f(x) = 6(5)^2 itself. This function, at first glance, might seem deceptively simple, but a closer examination reveals the underlying mathematical structure that governs its behavior. The function f(x) essentially defines a rule that takes an input, denoted by the variable x, and transforms it into an output. In this particular case, the rule involves a combination of multiplication and exponentiation. Let's break down the function into its key components:

  • The Constant Multiple: The function begins with a constant multiple of 6. This means that whatever the result of the rest of the expression is, it will be multiplied by 6. Constant multiples play a significant role in scaling functions, affecting their overall magnitude.
  • The Base and Exponent: The core of the function lies in the term (5)^2. This represents 5 raised to the power of 2, which is also known as 5 squared. Exponentiation signifies repeated multiplication, in this case, 5 multiplied by itself. The base, 5, is the number being multiplied, and the exponent, 2, indicates how many times the base is multiplied by itself.
  • The Absence of x: A notable characteristic of this function is the absence of the variable x within the expression. This implies that the function's output will be the same regardless of the input value of x. In other words, the function produces a constant output. This is a special case of a function known as a constant function. Understanding the absence of x is crucial for correctly evaluating the function for any given input.

By carefully dissecting the function into these components, we gain a clearer understanding of its inherent nature. This foundation will prove invaluable as we proceed to the next step: evaluating the function at x = -1.

Now that we have a firm grasp of the function f(x) = 6(5)^2, the next step in our quest to find f(-1) is the crucial process of substitution. Substitution, in the context of function evaluation, involves replacing the variable x in the function's expression with the specific value we are interested in, which in this case is -1. This step effectively transforms the general function f(x) into a specific expression that we can then evaluate arithmetically.

However, a key observation we made in the previous section comes into play here. We noted that the function f(x) = 6(5)^2 does not actually contain the variable x in its expression. This means that when we substitute x = -1, it doesn't directly appear anywhere within the expression. This might seem a bit perplexing at first, but it's a fundamental characteristic of constant functions.

Since there is no x to replace, the expression remains unchanged. Therefore, the substitution step, in this particular case, doesn't alter the expression. We are still left with 6(5)^2. This might seem like a trivial step, but it's an important realization. It reinforces the concept that for constant functions, the output is independent of the input. Regardless of what value we substitute for x, the function will always produce the same output.

In essence, the substitution step, in this context, highlights the nature of the function as a constant function. It sets the stage for the final step, where we will evaluate the expression 6(5)^2 to arrive at the final answer. This understanding is critical for accurately evaluating functions, especially those that might not explicitly involve the input variable.

Having successfully navigated the substitution step, we now arrive at the final stage of our journey: the calculation of 6(5)^2. This step involves applying the order of operations, a fundamental principle in mathematics that dictates the sequence in which operations should be performed to ensure consistent and accurate results. The order of operations is often remembered by the acronym PEMDAS, which stands for:

  • Parentheses
  • Exponents
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

In our expression, 6(5)^2, we have an exponent and a multiplication. According to PEMDAS, we must first address the exponent before performing the multiplication. Therefore, we begin by evaluating 5^2, which represents 5 raised to the power of 2, or 5 multiplied by itself.

5^2 = 5 * 5 = 25

Now that we have evaluated the exponent, our expression simplifies to 6(25). The remaining operation is multiplication. We multiply 6 by 25 to obtain the final result.

6 * 25 = 150

Therefore, the value of 6(5)^2 is 150. This result signifies the output of the function f(x) = 6(5)^2 for any input value of x, including x = -1. The systematic application of the order of operations ensures that we arrive at the correct and simplified exact value. This meticulous approach is crucial for accurate mathematical calculations and problem-solving.

In this comprehensive exploration, we embarked on a journey to unravel the intricacies of function evaluation, focusing on the function f(x) = 6(5)^2. Our primary goal was to determine the value of f(-1), a task that illuminated the core principles of function manipulation and substitution. We began by dissecting the function, understanding its structure and the role of each component. We identified the constant multiple of 6 and the exponential term (5)^2. A key observation was the absence of the variable x in the function's expression, indicating that it is a constant function.

Next, we proceeded to the crucial step of substitution, replacing the variable x with the specified value, -1. However, due to the absence of x in the expression, the substitution step didn't alter the expression. We were still left with 6(5)^2. This reinforced the concept that for constant functions, the output is independent of the input.

Finally, we meticulously performed the necessary arithmetic operations to evaluate 6(5)^2. We adhered to the order of operations (PEMDAS), first evaluating the exponent 5^2 to obtain 25, and then multiplying 6 by 25. This led us to the final result: 150.

Therefore, we can definitively conclude that f(-1) = 150 for the function f(x) = 6(5)^2. This value represents the simplified exact value of the function when x is equal to -1. This exploration not only provided us with a concrete answer but also deepened our understanding of function evaluation, constant functions, and the importance of the order of operations. The skills and knowledge gained here will undoubtedly prove invaluable in tackling a wide range of mathematical challenges involving functions and their applications.