Evaluate F(2) For F(x) = 2x² + 5√(x+2)
Introduction
Hey guys! Let's dive into a fun math problem today where we're going to evaluate a function. Specifically, we're given the function f(x) = 2x² + 5√(x+2) and our mission, should we choose to accept it (which we totally do!), is to find the value of f(2). This means we need to substitute '2' for 'x' in the function and simplify. Sounds like a plan? Awesome, let's get started!
This kind of problem is super common in algebra and calculus, so understanding how to tackle it is a fantastic skill to have in your mathematical toolkit. We'll break it down step-by-step, making sure every little detail is crystal clear. We'll talk about the order of operations, how to handle exponents, and how to deal with square roots. By the end of this, you'll be a pro at evaluating functions like this one, and you'll be able to confidently tackle similar problems in the future. So, grab your pencils (or your digital stylus!), and let's get this math party started! Remember, math isn't just about numbers and equations; it's about the journey of problem-solving, and we're in this together. Let's explore this fun mathematical landscape and conquer this problem with smiles on our faces!
Step-by-Step Evaluation of f(2)
Alright, let's get down to the nitty-gritty and actually evaluate f(2). Remember, our function is f(x) = 2x² + 5√(x+2). The first thing we need to do is substitute '2' for every 'x' we see in the function. This gives us:
f(2) = 2(2)² + 5√(2+2)
Now, we need to follow the order of operations, which you might remember as PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This is crucial to getting the right answer. First up, we have parentheses and exponents. Inside the square root, we have (2+2), which simplifies to 4. Also, we have (2)², which means 2 squared, or 2 times 2, which equals 4. Let's plug these values back into our equation:
f(2) = 2(4) + 5√4
Next, we deal with the square root. The square root of 4 is 2, because 2 times 2 equals 4. So, we can replace √4 with 2:
f(2) = 2(4) + 5(2)
Now we're onto multiplication. We have 2(4), which is 8, and 5(2), which is 10. Our equation now looks like this:
f(2) = 8 + 10
Finally, we perform the addition. 8 plus 10 equals 18. So, we have:
f(2) = 18
And there you have it! We've successfully evaluated f(2). It turns out that when x is 2, the function f(x) equals 18. That wasn't so bad, was it? We just followed the order of operations, step by step, and we arrived at the solution. Remember, breaking down a problem into smaller, manageable steps is a key strategy in mathematics (and in life, really!). So, let's celebrate this small victory and move on to discussing why this process is so important.
The Importance of Evaluating Functions
Okay, so we've figured out that f(2) = 18 for our function. But you might be wondering, “Why bother? What's the big deal about evaluating functions anyway?” That's a totally valid question, and it's important to understand the why behind the math, not just the how. Evaluating functions is a fundamental concept in mathematics, and it has tons of real-world applications. Think of it like this: functions are like little machines. You feed them an input (in our case, the number 2), and they spit out an output (which turned out to be 18).
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