Evaluating Functions Finding G(3) For G(x) = X^2 + 12x

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Hey guys! Let's dive into the world of functions today. We're going to break down how to evaluate a function, using the example g(x) = x^2 + 12x. Don't worry, it's not as intimidating as it looks! We'll take it step by step and by the end, you'll be a function-evaluating pro. Evaluating functions is a fundamental concept in mathematics, and it's super important for understanding more advanced topics later on. So, buckle up and let's get started!

Understanding Functions

Before we jump into evaluating, let's make sure we're all on the same page about what a function actually is. In its simplest form, a function is like a machine. You feed it an input (we often call this 'x'), and it spits out an output (which we often call 'g(x)' or 'y'). The function has a specific rule or set of rules that it follows to transform the input into the output. Think of it like a vending machine: you put in money (the input), and you get a snack (the output). The machine has a set process for determining which snack you get based on your money.

In our case, the function is g(x) = x^2 + 12x. This means that whatever value we put in for 'x', the function will first square it (x^2) and then add 12 times the original value of 'x' (12x). The result of this calculation is the output, g(x). Functions are the bedrock of so much of mathematics, from algebra and calculus to more advanced fields. They allow us to model relationships between different quantities and make predictions. Mastering the basics of functions, like evaluating them, is crucial for success in math. So, don't be afraid to ask questions and practice as much as you can!

Let's break down the notation a little more. The 'g' in g(x) is just the name of the function. We could have called it 'f', 'h', or anything else. The '(x)' part tells us that 'x' is the input variable. And the expression 'x^2 + 12x' is the rule that the function follows. When we evaluate a function, we're essentially asking: "What happens if I put this specific value in for 'x'?" The answer is the output, or the value of the function at that point. So, keep this in mind as we go through the examples – we're simply substituting values into the function's rule to find the corresponding output.

Evaluating g(3) for g(x) = x^2 + 12x

Now, let's get to the heart of the matter: evaluating g(3) when g(x) = x^2 + 12x. This means we're going to substitute '3' for every 'x' in the function's expression. It's like we're telling the function machine, "Hey, process the number 3 for me!" Remember, the function's rule is to square the input and then add 12 times the input. So, we're going to do exactly that with the number 3. This process of substitution is key to evaluating any function. You're simply replacing the variable with the given value and then simplifying the expression. It's a straightforward process, but it's crucial to pay attention to the order of operations (PEMDAS/BODMAS) to avoid mistakes.

Here's how we do it step-by-step:

  1. Write down the function: g(x) = x^2 + 12x
  2. Substitute 3 for x: g(3) = (3)^2 + 12(3)
  3. Evaluate the exponent: g(3) = 9 + 12(3)
  4. Perform the multiplication: g(3) = 9 + 36
  5. Add the numbers: g(3) = 45

So, g(3) = 45. This means that when we input 3 into the function g(x), the output is 45. That's it! We've successfully evaluated the function at x = 3. It might seem simple, but this is a foundational skill that you'll use again and again in math. The key is to be careful with your substitutions and follow the order of operations. Practice makes perfect, so let's try another example to solidify your understanding. Don't hesitate to go back over these steps if you need a refresher.

Key Takeaways

  • Functions are machines: They take an input and produce an output based on a rule.
  • Evaluating means substituting: Replace the variable 'x' with the given value.
  • Follow the order of operations: PEMDAS/BODMAS is your friend.
  • Practice makes perfect: The more you evaluate functions, the easier it becomes.

Evaluating functions might seem like a small step, but it's a crucial building block for more advanced mathematical concepts. By understanding how functions work and how to evaluate them, you're setting yourself up for success in future math courses. Keep practicing, and don't be afraid to ask questions. You've got this!