Evaluating F(3) Given F(x) = 5 + 4x^2

Introduction

In this article, we will delve into the process of evaluating a function at a specific point. Specifically, we'll explore how to find the value of f(3) given the function f(x) = 5 + 4x^2. This is a fundamental concept in mathematics and is crucial for understanding function behavior and applications in various fields. Understanding how to evaluate functions is a crucial skill in mathematics. A function, in simple terms, is a rule that takes an input value, often represented by x, and produces an output value. This output is typically denoted as f(x), read as "f of x." The expression f(x) = 5 + 4x^2 defines a particular function where the output is determined by squaring the input x, multiplying it by 4, and then adding 5. Evaluating a function means finding the output value f(x) for a given input value x. This process is fundamental to understanding the behavior of functions and their applications in various fields, including algebra, calculus, and real-world modeling. Our focus here is on finding f(3), which means we need to determine the output of this function when the input is 3. This involves substituting 3 for x in the function's expression and then simplifying the resulting arithmetic expression. The steps we'll follow are straightforward but illustrate the core concept of function evaluation. This skill is essential for analyzing functions, solving equations, and understanding the relationships between variables in mathematical models. We will break down each step in detail to ensure a clear understanding of the process. This includes explaining the order of operations, the substitution process, and the simplification of the expression. By the end of this article, you should be able to confidently evaluate similar functions for any given input value. Furthermore, understanding function evaluation lays the groundwork for more advanced mathematical concepts, such as graphing functions, finding their derivatives and integrals, and applying them to solve practical problems. So, let's begin our journey into the world of function evaluation and discover how to find the value of f(3) for the given function.

Understanding the Function f(x) = 5 + 4x^2

Before we dive into evaluating f(3), let's first understand the function itself. The function f(x) = 5 + 4x^2 is a quadratic function. Quadratic functions are characterized by the presence of a term where the variable x is raised to the power of 2. This x^2 term gives the function its parabolic shape when graphed. The other components of the function, the constant 5 and the term 4x^2, influence the parabola's position and shape on the coordinate plane. The constant term, 5, represents the vertical shift of the parabola. In this case, it shifts the parabola 5 units upward along the y-axis. The coefficient of the x^2 term, which is 4, determines the parabola's vertical stretch or compression. A larger coefficient results in a narrower parabola, while a smaller coefficient results in a wider parabola. In this case, the 4 stretches the parabola vertically. To fully grasp how this function works, imagine plugging in different values for x. For each x you input, the function will perform a specific calculation: it will square the x, multiply the result by 4, and then add 5. The result of this calculation is the function's output, f(x). This function evaluation process is the core of understanding how functions behave. By understanding the function's components and how they interact, we can predict the function's output for any given input. This is crucial for various mathematical tasks, such as solving equations, graphing functions, and modeling real-world phenomena. The f(x) = 5 + 4x^2 function, despite its simple form, encapsulates many fundamental concepts of functions in mathematics. Understanding its structure and behavior provides a solid foundation for exploring more complex functions and their applications. In the following sections, we will focus on the specific task of evaluating this function at x = 3, but the underlying principles apply to evaluating any function at any given point. This involves substituting the input value into the function's expression and simplifying the result, a process that we will break down step-by-step to ensure clarity and understanding.

Step-by-Step Evaluation of f(3)

To evaluate f(3), we need to substitute x = 3 into the function f(x) = 5 + 4x^2. This means replacing every instance of x in the function's expression with the value 3. Let's break down the process step-by-step: The first step is the substitution. We replace x with 3 in the expression 5 + 4x^2, resulting in 5 + 4(3)^2. It's crucial to use parentheses around the 3 to ensure that the squaring operation is performed correctly before the multiplication. Next, we follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). According to PEMDAS, we must first address the exponent. We have (3)^2, which means 3 squared, or 3 multiplied by itself. Thus, (3)^2 = 3 * 3 = 9. Our expression now becomes 5 + 4(9). The next operation in PEMDAS is multiplication. We have 4(9), which means 4 multiplied by 9. This gives us 4 * 9 = 36. Our expression is now simplified to 5 + 36. Finally, we perform the addition. We add 5 and 36, which gives us 5 + 36 = 41. Therefore, f(3) = 41. This means that when the input to the function f(x) = 5 + 4x^2 is 3, the output is 41. This step-by-step evaluation demonstrates the process of finding the value of a function at a specific point. It highlights the importance of following the order of operations to ensure accurate results. Each step, from substitution to simplification, is crucial for arriving at the correct answer. This process of function evaluation is a fundamental skill in mathematics and is used extensively in various applications. Understanding how to evaluate functions allows us to analyze their behavior, solve equations, and model real-world phenomena. In the next section, we will summarize our findings and reiterate the key steps involved in evaluating functions.

Conclusion: The Value of f(3) and the Importance of Function Evaluation

In conclusion, we have successfully evaluated the function f(x) = 5 + 4x^2 at x = 3. By substituting 3 for x and following the order of operations, we found that f(3) = 41. This result tells us the output of the function when the input is 3. The process of function evaluation is a cornerstone of mathematics. It allows us to understand how functions behave and to make predictions about their outputs for different inputs. This is essential for solving equations, graphing functions, and applying mathematical models to real-world problems. The steps involved in function evaluation are straightforward but crucial: substitution and simplification. First, we substitute the given input value for the variable in the function's expression. Then, we simplify the resulting expression by following the order of operations. This systematic approach ensures that we arrive at the correct answer. Understanding function evaluation is not just about finding a numerical result; it's about gaining insight into the function's behavior. By evaluating a function at different points, we can begin to understand its graph, its rate of change, and its overall properties. This knowledge is invaluable in various fields, including physics, engineering, economics, and computer science. The function f(x) = 5 + 4x^2 is a simple example, but the principles of function evaluation apply to all types of functions, from simple linear functions to complex trigonometric and exponential functions. Mastering this skill is essential for success in higher-level mathematics and its applications. Therefore, understanding and practicing function evaluation is a crucial step in your mathematical journey. It provides the foundation for more advanced concepts and empowers you to solve a wide range of problems. Remember the steps, understand the process, and you'll be well-equipped to tackle any function evaluation challenge.