Evaluate F(0) For F(x) = 2x² + 5√(x+2)

Introduction

Hey guys! Today, we're diving into a fun little math problem where we need to figure out the value of a function at a specific point. Specifically, we've got the function f(x) = 2x² + 5√(x+2), and our mission, should we choose to accept it (and we totally do!), is to find out what f(0) is. Don't worry, it's not as daunting as it might look at first glance. We're going to break it down step by step, so even if you're not a math whiz, you'll be able to follow along and maybe even impress your friends with your newfound function-evaluating skills. So, grab your calculators (or just your brain, if you're feeling particularly sharp!), and let's get started!

Understanding the Function

Before we jump into plugging in numbers, let's take a moment to really understand what this function, f(x) = 2x² + 5√(x+2), is all about. Functions, in general, are like little machines: you feed them an input (in this case, a value for x), and they spit out an output (the value of f(x)). Our function here has two main parts: a quadratic term (2x²) and a square root term (5√(x+2)). The quadratic part means we're squaring x and then multiplying it by 2. The square root part means we're adding 2 to x, taking the square root of the result, and then multiplying by 5. Each of these parts contributes to the overall behavior of the function, and understanding them helps us predict what kind of output we might expect. For instance, the square root part reminds us that we need to be careful about negative numbers inside the square root, as that would give us imaginary numbers (which we're not dealing with today). Now that we've got a good grasp of the function's anatomy, we're ready to tackle the main question: What happens when x is 0?

Step-by-Step Evaluation

Alright, let's get down to the nitty-gritty! To find f(0), we're going to do exactly what it sounds like: replace every x in the function's formula with a 0. This is a fundamental concept in algebra, and once you get the hang of it, you'll be evaluating functions like a pro. So, let's take f(x) = 2x² + 5√(x+2) and swap out those x's with 0's. We get:

f(0) = 2(0)² + 5√(0+2)

Now, it's just a matter of following the order of operations (PEMDAS/BODMAS, remember?) to simplify this expression. First up, we handle the exponents. 0 squared (0²) is simply 0, so we have:

f(0) = 2(0) + 5√(0+2)

Next, we take care of the multiplication. 2 times 0 is 0, so our equation becomes:

f(0) = 0 + 5√(0+2)

Now, let's tackle the stuff inside the square root. 0 plus 2 is 2, giving us:

f(0) = 0 + 5√2

And finally, we deal with the square root. The square root of 2 is an irrational number, meaning it goes on forever without repeating. But for our purposes, we can use a calculator to get an approximate value, which is roughly 1.414. So, we have:

f(0) = 0 + 5(1.414)

One last multiplication step: 5 times 1.414 is approximately 7.07. Therefore:

f(0) = 0 + 7.07

Adding 0 doesn't change anything, so we're left with:

f(0) = 7.07

So, there you have it! f(0) is approximately 7.07.

Rounding to the Nearest Hundredth

Our instructions specifically tell us to round our answer to the nearest hundredth. Luckily, we've already done that! The hundredths place is the second digit after the decimal point, and in our case, it's 7. So, 7.07 is already rounded to the nearest hundredth. If we had a number like 7.075, we would round it up to 7.08, but we don't need to worry about that here. This rounding step is crucial in many mathematical and scientific contexts, as it ensures we're presenting our results with the appropriate level of precision. It's a small detail, but it can make a big difference in the accuracy of our work. So, always remember to pay attention to rounding instructions!

Final Answer: f(0) ≈ 7.07

So, after carefully evaluating the function f(x) = 2x² + 5√(x+2) at x = 0, we've arrived at our final answer: f(0) is approximately equal to 7.07. We broke down the problem step by step, making sure to follow the order of operations and handle each term with care. We also remembered to round our answer to the nearest hundredth, as the question instructed. This is a great example of how breaking down a complex problem into smaller, more manageable steps can make it much easier to solve. And remember, practice makes perfect! The more you work with functions and function evaluation, the more comfortable and confident you'll become. Great job, guys! You've successfully navigated this math challenge, and you're one step closer to becoming a function-evaluating master!

Discussion Category: Mathematics

This problem falls squarely into the mathematics category. Specifically, it's a problem related to function evaluation, which is a fundamental concept in algebra and calculus. It involves understanding the definition of a function, substituting a given value into the function's formula, and simplifying the resulting expression. Function evaluation is a crucial skill in many areas of mathematics and its applications, so mastering it is definitely a worthwhile endeavor. If you're looking for more practice with function evaluation, you can explore other examples, try different types of functions (like polynomials, trigonometric functions, or exponential functions), and challenge yourself with more complex expressions. Keep up the great work, and you'll be a math whiz in no time!