Evaluating F(0) For F(x) = 2(x)^2+5√(x+2) - A Step-by-Step Solution

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Understanding Function Evaluation

In mathematics, a function is a relationship between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. Evaluating a function means substituting a specific value for the input variable (often denoted as 'x') and calculating the corresponding output. This process allows us to understand the behavior of the function at different points and to explore its properties. In the given problem, we are presented with the function f(x) = 2x² + 5√(x+2), and our task is to evaluate this function at x = 0. This involves replacing every instance of 'x' in the function's expression with the value '0' and then simplifying the resulting expression using the order of operations. Understanding the concept of function evaluation is crucial in various areas of mathematics and its applications, including calculus, algebra, and mathematical modeling.

Step-by-Step Calculation of f(0)

To find the value of f(0), we substitute x = 0 into the function's equation: f(x) = 2x² + 5√(x+2). This gives us f(0) = 2(0)² + 5√(0+2). Now, we simplify the expression following the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). First, we evaluate the exponent: 0² = 0. Next, we simplify the expression inside the square root: 0 + 2 = 2. Our equation now looks like this: f(0) = 2(0) + 5√2. Next, we perform the multiplication: 2(0) = 0. So, we have f(0) = 0 + 5√2. Finally, we need to calculate the value of 5√2. The square root of 2 is approximately 1.41421. Multiplying this by 5, we get 5√2 ≈ 5 * 1.41421 ≈ 7.07105. Therefore, f(0) ≈ 0 + 7.07105 ≈ 7.07105. Rounding this to the nearest hundredth, we get f(0) ≈ 7.07.

Detailed Breakdown of the Process

Let's break down the calculation of f(0) step by step to ensure clarity. We begin with the function f(x) = 2x² + 5√(x+2). Our objective is to find the value of this function when x = 0. The first step is to substitute x with 0 in the equation: f(0) = 2(0)² + 5√(0+2). Now, we simplify each part of the expression. First, consider the term 2(0)². The square of 0 is 0, so we have 2(0) = 0. Next, we look at the term 5√(0+2). Inside the square root, we have 0 + 2, which simplifies to 2. So, the term becomes 5√2. The square root of 2 is an irrational number, approximately equal to 1.41421. We then multiply this value by 5: 5 * 1.41421 ≈ 7.07105. Now, we combine the two simplified terms: f(0) = 0 + 7.07105. This gives us f(0) ≈ 7.07105. Finally, we round this value to the nearest hundredth. The hundredth place is the second digit after the decimal point. In this case, it's the 7 in 7.07. The digit after the 7 is a 1, which is less than 5, so we round down and keep the 7. Therefore, f(0) ≈ 7.07.

Practical Applications of Function Evaluation

Function evaluation is not just a mathematical exercise; it has numerous practical applications in various fields. In physics, for example, functions are used to model the motion of objects, and evaluating the function at a specific time gives the position or velocity of the object at that time. In economics, functions can represent cost, revenue, or profit, and evaluating these functions can help businesses make informed decisions. In computer science, functions are the building blocks of programs, and evaluating them is essential for executing code. Furthermore, in data analysis and machine learning, functions are used to model relationships between variables, and evaluating these functions is crucial for making predictions and understanding patterns. The ability to evaluate functions accurately and efficiently is therefore a fundamental skill in many STEM disciplines and professional contexts. Understanding how a function behaves at different input values allows us to make predictions, optimize processes, and solve real-world problems.

Conclusion: The Value of f(0) and its Significance

In conclusion, by substituting x = 0 into the function f(x) = 2x² + 5√(x+2) and simplifying, we found that f(0) ≈ 7.07 when rounded to the nearest hundredth. This process of function evaluation is a fundamental concept in mathematics and has wide-ranging applications in various fields. Understanding how to evaluate functions allows us to analyze mathematical models, make predictions, and solve real-world problems. The value of f(0) represents the output of the function at the specific input x = 0, providing a snapshot of the function's behavior at that point. This single value can be crucial in understanding the overall characteristics of the function, especially in the context of its domain and range. Mastering function evaluation is therefore an essential skill for anyone pursuing studies or careers in mathematics, science, engineering, or technology.

Keywords and Rephrased Question

Repair-input-keyword: Given the function f(x) = 2x² + 5√(x+2), determine the value of f(0), rounding your answer to the nearest hundredth.

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