Hey guys! Today, we're diving into the world of functions, specifically how to evaluate them. We've got a function here, f(x) = 3x² + 2x - 5, and our mission, should we choose to accept it (and we do!), is to find the value of this function at a couple of different inputs. Think of it like this: a function is like a machine, you feed it a number, and it spits out another number based on its internal rules. In our case, the rule is 3x² + 2x - 5. Let's crack this open and see what we get!
Understanding Function Evaluation
Before we jump into the specific calculations, let's make sure we're all on the same page about what function evaluation actually means. Essentially, when we see something like f(2), it's asking us: "What is the value of the function f when we replace x with 2?" So, everywhere we see an x in the function's formula, we're going to substitute it with the given input value. This might sound a bit abstract, but it's super straightforward once you've done a few examples, and I promise, we're about to do just that! Imagine you have a recipe for a cake, and the recipe tells you how much of each ingredient you need. The function is like the recipe, and the input is like saying, "Okay, let's make a cake that's twice as big!" You'd need to adjust all the ingredients accordingly, right? Function evaluation is the same idea – we're adjusting the function's output based on the input we provide. The key is to be meticulous and follow the order of operations (PEMDAS/BODMAS) to ensure we get the correct answer. We'll be focusing on that as we work through the problems. Remember, practice makes perfect, and by the end of this article, you'll be a function evaluation pro! This concept is fundamental to many areas of mathematics, from calculus to algebra, so mastering it now will set you up for success in your future math endeavors. So, buckle up, and let's get started!
(a) Finding f(2)
Okay, let's tackle the first part: finding f(2). This means we're going to substitute every x in our function f(x) = 3x² + 2x - 5 with the number 2. So, let's rewrite the function with the substitution:
f(2) = 3(2)² + 2(2) - 5
Now, we need to follow the order of operations (PEMDAS/BODMAS). First up, we handle the exponent: 2² is 2 multiplied by itself, which is 4. So, we now have:
f(2) = 3(4) + 2(2) - 5
Next, we perform the multiplications. 3 multiplied by 4 is 12, and 2 multiplied by 2 is 4. Our equation now looks like this:
f(2) = 12 + 4 - 5
Finally, we do the addition and subtraction from left to right. 12 plus 4 is 16, and 16 minus 5 is 11. So, we've arrived at our answer:
f(2) = 11
There you have it! When we plug in 2 into our function, we get 11. It's like our function machine took the number 2 and transformed it into 11. Pretty neat, huh? Now, before we move on, let's just take a quick look back at the steps we took. We substituted, we dealt with the exponent, we multiplied, and then we added and subtracted. This is the general process for evaluating functions, and if you follow these steps carefully, you'll be golden every time. Notice how important it is to follow the order of operations. If we had, for example, added 4 and -5 before multiplying, we would have gotten the wrong answer. That's why remembering PEMDAS/BODMAS is crucial. Now that we've conquered f(2), let's move on to the next challenge!
(b) Finding f( ): A Placeholder Mystery
Alright, guys, this is where things get a little… interesting. We've got f( ), but instead of a number inside the parentheses, we have a circle. What does this mean? Well, in this context, the circle is likely a placeholder. It's telling us that we need to substitute whatever expression is represented by that circle into our function. Without knowing what the circle represents, we can't get a numerical answer like we did with f(2). Instead, we'll end up with an expression that involves whatever the circle stands for. Think of it like this: imagine the circle is a box, and we don't know what's inside the box yet. We can still write down a recipe for what to do with the contents of the box, even if we don't know exactly what those contents are. So, let's say the circle represents some expression, let's call it "expression" for now. We're going to substitute "expression" for x in our function f(x) = 3x² + 2x - 5. This will give us a general formula for f of anything! This is a powerful idea in mathematics – the ability to work with abstract expressions and manipulate them to reveal underlying relationships. It's like being able to write a computer program that works for any input, not just a specific one. This kind of thinking is what separates algebra from arithmetic, and it's a crucial step in your mathematical journey. So, let's embrace the mystery of the circle and see where it leads us!
Substituting the Placeholder
Let's perform the substitution. We replace every x in f(x) = 3x² + 2x - 5 with our placeholder "expression":
f(expression) = 3(expression)² + 2(expression) - 5
And that's it! We've evaluated the function f at the placeholder "expression". We can't simplify this any further without knowing what "expression" actually is. This might seem a bit unsatisfying, especially after getting a nice, neat numerical answer for f(2). But this is a crucial step in understanding how functions work. We've created a general formula that will work for any input, not just a specific number. Think of it like this: we've built a machine that can process any ingredient, not just one particular ingredient. This flexibility is what makes functions so powerful in mathematics. Now, let's consider a few possibilities. What if the circle represented another variable, like y? Then our answer would be f(y) = 3y² + 2y - 5. What if the circle represented a whole other function, like g(x)? Then our answer would be f(g(x)) = 3(g(x))² + 2(g(x)) - 5. This is called function composition, and it's a fundamental concept in calculus. The point is, by leaving the circle as a placeholder, we've created a very versatile result. We can plug in anything we want, and our formula will tell us the output. This is the beauty of abstraction in mathematics. We've taken a specific problem and generalized it to a much wider class of problems. So, even though we don't have a numerical answer, we've actually gained a deeper understanding of the function f. We've seen how it acts on any input, not just the number 2.
Key Takeaways and Next Steps
So, guys, what have we learned today? We've explored the process of function evaluation, both with a specific numerical input and with a placeholder. We've seen how to substitute values into a function and how to follow the order of operations to get the correct answer. We've also encountered the idea of a placeholder, which allows us to create general formulas that work for any input. These are fundamental concepts in mathematics, and mastering them will set you up for success in more advanced topics. Remember, practice is key! The more you work with functions, the more comfortable you'll become with evaluating them. Try plugging in different numbers into our function f(x) = 3x² + 2x - 5. What happens when you plug in a negative number? What happens when you plug in zero? What happens when you plug in a fraction? These are all great questions to explore. And don't be afraid to experiment with other functions as well. There are countless functions out there, each with its own unique behavior. The more you explore, the better you'll understand how functions work. Also, think about how functions are used in the real world. They're used to model everything from the trajectory of a baseball to the growth of a population. Understanding functions is essential for understanding the world around us. So, keep practicing, keep exploring, and keep asking questions. And most importantly, have fun! Math can be challenging, but it can also be incredibly rewarding. With a little effort and a lot of curiosity, you can unlock the secrets of the mathematical universe. Now, go forth and evaluate some functions!
If you're looking to take your understanding further, I recommend exploring function composition, inverse functions, and different types of functions like linear, quadratic, and exponential functions. These topics build upon the foundation we've laid today and will open up a whole new world of mathematical possibilities. And remember, there are tons of resources available online and in textbooks to help you along the way. Don't hesitate to reach out for help if you get stuck. There's a whole community of mathematicians out there who are eager to share their knowledge and passion for the subject. So, keep learning, keep growing, and keep exploring the wonderful world of mathematics!