Calculating (g O F)(2): A Step-by-Step Guide

Hey guys! Today, we're diving into a math problem that involves function composition. Specifically, we're going to figure out the value of (g o f)(2). Don't worry if the notation looks a little intimidating; it's actually pretty straightforward. We'll break it down step-by-step so you can easily understand how to solve this. The core concept here is function composition, which means applying one function to the result of another function. We're given two functions: f(x) and g(x). Our goal is to find (g o f)(2), which means we first apply the function f to the input 2, and then apply the function g to the result of f(2). This might sound complicated, but trust me, it's not. It's like a recipe where you need to follow the steps in a specific order. So, let's get started and break down this problem. We'll go through each step carefully, explaining the process along the way. By the end of this, you'll be able to solve similar problems with confidence. Let's make sure we understand the functions and the composition operation. Remember, understanding the basics is always the key to solving more complex problems. It's really about substituting values into the equations and simplifying them. This process will enable us to master the function composition, making us better in algebra. So, let's dive into the calculation to clarify everything.

Understanding the Functions: f(x) and g(x)

Alright, before we get to the core of the problem, let's make sure we understand the functions we are working with. We are given two functions: f(x) = 3x + 2 and g(x) = x² - 5x + 3. In simple terms, a function is like a machine that takes an input (x in this case), does something to it (according to the formula), and gives you an output. The function f(x) takes an input, multiplies it by 3, and then adds 2. For instance, if you put x = 1 into f(x), you get f(1) = 3*1 + 2 = 5. The function g(x) is a bit more complex since it involves squaring the input, multiplying it by 5, and then adding 3. For example, if you put x = 2 into g(x), you get g(2) = 2² - 5*2 + 3 = 4 - 10 + 3 = -3. Understanding what each function does is crucial because, in function composition, we use the output of one function as the input for the other. So, knowing how each one works helps us predict what to expect. We'll be using these functions together to find the value of (g o f)(2). Remember, the key is to perform the steps in the correct order: first, find the output of f(2), and then use that output as the input for g. This step prepares us to tackle the main composition problem. So, understanding these individual functions sets the foundation for our next calculations. It makes the entire process clear.

Step-by-Step Calculation of (g o f)(2)

Now, let's get down to brass tacks and find the value of (g o f)(2). As we mentioned earlier, function composition involves applying one function to the result of another. In this case, we first need to find f(2). We do this by substituting x = 2 into the function f(x). So, f(2) = 3*2 + 2 = 6 + 2 = 8. Great! We now know that f(2) = 8. The next step is to use this result as the input for the function g(x). This means we need to find g(8). We substitute x = 8 into the function g(x). So, g(8) = 8² - 5*8 + 3 = 64 - 40 + 3 = 27. Therefore, (g o f)(2) = g(f(2)) = g(8) = 27. Voila! We have successfully calculated the value of (g o f)(2). The solution is 27. The core idea is that we performed the functions in the correct order. First, we got the output of f(2), then used that output as an input in the g(x) function. By taking it step by step, it breaks down into a manageable problem. This approach helps us solve the problem systematically, and prevents any errors. Remember that function composition may sound complex, but with practice and a good understanding of functions, it becomes easy to solve. So, now you know how to compute (g o f)(2). You can apply this same approach to other composite functions as well. Always start from the inside function and work your way out.

Conclusion and Final Answer

Alright, guys, let's wrap things up! We've successfully calculated the value of (g o f)(2) by following a clear, step-by-step approach. We started by understanding what the functions f(x) and g(x) do individually. Then, we found f(2), which turned out to be 8. After that, we used this result as the input for the function g(x), resulting in g(8) = 27. Therefore, (g o f)(2) = 27. The important takeaway is the process: always work from the inner function outward. First, evaluate the inner function with the given input. Next, use the result as the input for the outer function. Keep practicing this method, and you'll become proficient in function composition in no time! So, the final answer to the question, (g o f)(2) = ? is 27. Congratulations, you've conquered this math problem! Now, you're ready to tackle similar problems. Feel free to try out some more examples to solidify your understanding. Keep in mind that the key is to break down the process into small, manageable steps. Practice makes perfect, and with each practice, you will become more comfortable with this math concept.