Evaluating Composite Functions: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of composite functions. Specifically, we're going to break down how to evaluate $(f \circ g)(0)$ when given $f(x) = 3x + 5$ and $g(x) = 3x^2 - 7$. Composite functions might sound intimidating, but don't worry, we'll take it one step at a time and you'll be a pro in no time. Let's get started!

Understanding Composite Functions

Before we jump into the problem, let's make sure we're all on the same page about what a composite function actually is. Think of it like a function within a function. The notation $(f \circ g)(x)$ means we're taking the function $g(x)$ and plugging it into the function $f(x)$. In other words, we first evaluate $g(x)$, and then we take that result and use it as the input for $f(x)$. It's like a two-step process, where the output of one function becomes the input of another. So, when you see $(f \circ g)(x)$, read it as "f of g of x". This little understanding is crucial because it dictates the order in which we perform the operations. We always work from the inside out. This means we evaluate the inner function, $g(x)$ in this case, first, and then use its result to evaluate the outer function, $f(x)$. Mastering this concept will make evaluating any composite function a breeze. Remember, it's all about the order of operations and understanding how one function's output feeds into another. We'll see this in action as we solve our example problem, and you'll see just how straightforward it can be once you grasp the core idea. So, let's keep this in mind as we move forward: inside out, always!

Step 1: Evaluate the Inner Function g(0)

The first thing we need to do, as we discussed, is to evaluate the inner function, which is $g(x)$. In our case, we want to find $g(0)$. Remember that $g(x) = 3x^2 - 7$. So, to find $g(0)$, we simply substitute $x$ with $0$ in the expression for $g(x)$. This gives us $g(0) = 3(0)^2 - 7$. Now, let's simplify this. First, we calculate $0^2$, which is just $0$. Then, we multiply that by $3$, which gives us $3 * 0 = 0$. Finally, we subtract $7$ from $0$, resulting in $0 - 7 = -7$. Therefore, we've found that $g(0) = -7$. This is a key intermediate result because it's the value we'll use as the input for our next step. Make sure you're comfortable with this substitution process – it's a fundamental skill when working with functions. We're essentially figuring out what $g(x)$ outputs when we feed it the input $0$. This output, $-7$, is now ready to be used as the input for the outer function, $f(x)$. So, with $g(0)$ successfully evaluated, we're perfectly set up to move on to the next part of our composite function journey. Remember, we're building this step-by-step, and each step relies on the one before it.

Step 2: Evaluate the Outer Function f(g(0))

Now that we know $g(0) = -7$, we can move on to the second part of evaluating our composite function, $(f \circ g)(0)$. Remember, this means we need to find $f(g(0))$, and since we just found that $g(0) = -7$, we can rewrite this as $f(-7)$. So, now we're dealing with the function $f(x) = 3x + 5$, and we need to find its value when $x$ is $-7$. Just like before, we substitute the value of $x$ into the expression for $f(x)$. This means we replace $x$ with $-7$, giving us $f(-7) = 3(-7) + 5$. Next, we simplify this expression. First, we multiply $3$ by $-7$, which gives us $-21$. Then, we add $5$ to $-21$, resulting in $-21 + 5 = -16$. Therefore, we've found that $f(-7) = -16$. This is the final step in our evaluation process! We've successfully taken the output of $g(0)$ and used it as the input for $f(x)$, and we've calculated the resulting output. This whole process highlights the beauty of composite functions – how they chain together to create new and interesting relationships between inputs and outputs. So, feel proud of yourself for reaching this point! You've navigated the substitution, the arithmetic, and the overall concept of composite functions. We're almost at the finish line, but before we celebrate, let's state our final answer clearly.

Step 3: State the Final Answer

After all our hard work, we've reached the final step: stating the answer! We've successfully navigated the process of evaluating the composite function $(f \circ g)(0)$. We found that $g(0) = -7$, and then we used that result to find $f(g(0)) = f(-7) = -16$. Therefore, the final answer is $(f \circ g)(0) = -16$. Congratulations! You've made it through the entire process. It's essential to clearly state your final answer so there's no ambiguity. This is especially important in math, where precision is key. Think of it as the grand finale of your calculation – the moment you present the result of your efforts. Now, let's take a moment to reflect on what we've done. We started with the concept of composite functions, broke down the notation, and then meticulously worked through the steps of evaluating $(f \circ g)(0)$. We evaluated the inner function first, then used that output as the input for the outer function, and finally arrived at our answer. This systematic approach is what makes math problems solvable, even when they seem complex at first glance. So, pat yourself on the back for sticking with it and reaching the solution! You've not only found the answer but also reinforced your understanding of composite functions. And that's a victory worth celebrating.

Conclusion

So, there you have it! We've successfully evaluated the composite function $(f \circ g)(0)$ given $f(x) = 3x + 5$ and $g(x) = 3x^2 - 7$. We've seen how to break down the problem into manageable steps, starting with the inner function and working our way outwards. Remember, the key to mastering composite functions is understanding the order of operations and the concept of one function's output becoming another's input. This example provides a solid foundation for tackling other composite function problems, and with a little practice, you'll be evaluating them like a pro. Don't be afraid to take things slow, review the steps if needed, and most importantly, believe in your ability to solve these problems. Math can be challenging, but with a clear approach and a bit of perseverance, you can conquer any function that comes your way. Keep practicing, keep learning, and most importantly, keep enjoying the process of problem-solving. You've got this! And now, armed with this knowledge, you're ready to tackle more mathematical adventures. Go forth and calculate!