Solve F(g(-2)) For Two Functions

Hey mathletes! Today, we're diving into the awesome world of function composition. It might sound a bit fancy, but trust me, guys, it's super cool once you get the hang of it. We're going to tackle a problem involving two functions, $f(x)=-\frac{1}{2} x^2+5 x$ and $g(x)=x^2+2$, and figure out the value of $f(g(-2))$. So, grab your calculators, your pencils, and let's get this done!

Understanding Function Composition

First off, what exactly is $f(g(-2))$? In simple terms, it means we first figure out what $g(-2)$ is, and then we take that result and plug it into our function $f(x)$. Think of it like a relay race where the output of one function becomes the input for the next. It's all about building step-by-step. We have our two functions, $f(x)$ and $g(x)$. Our mission, should we choose to accept it, is to find the value of $f$ evaluated at the output of $g$ when the input is $-2$. This is a fundamental concept in algebra, and mastering it will unlock a whole new level of understanding how functions work together. It's not just about crunching numbers; it's about understanding the flow of information through mathematical expressions. When we talk about $f(g(x))$, we're essentially creating a new, composite function. This new function takes an input, passes it through $g$, and then passes the result of $g$ through $f$. In our specific case, the input is a fixed number, $-2$. So, we're not looking for a general formula for $f(g(x))$, but a specific numerical value. This often makes the problem a bit more straightforward, as we're dealing with concrete numbers rather than abstract variables. The functions themselves, $f(x)=-\frac{1}{2} x^2+5 x$ and $g(x)=x^2+2$, are both quadratic functions. This means their graphs are parabolas. $g(x)$ is an upward-opening parabola, while $f(x)$ is a downward-opening parabola. The composition of two quadratics can result in a polynomial of degree four, but since we're evaluating at a specific point, we'll avoid that complexity for now. The key takeaway here is the order of operations: evaluate the inner function first, then the outer function. It's like peeling an onion, layer by layer, to get to the core value.

Step 1: Evaluate the Inner Function, $g(-2)$

Our first move is to figure out the value of $g(-2)$. Remember, our function $g(x)$ is defined as $g(x)=x^2+2$. To find $g(-2)$, we simply substitute $-2$ for every $x$ in the expression for $g(x)$. So, we have $g(-2) = (-2)^2 + 2$. Now, let's do the math, guys. The square of $-2$ is $(-2) imes (-2)$, which equals $4$. So, $g(-2) = 4 + 2$. And $4 + 2$ is, you guessed it, $6$. Therefore, $g(-2) = 6$. This is a crucial first step, and it's important to be careful with the signs, especially when squaring negative numbers. A common mistake here is accidentally getting a negative result for $(-2)^2$, which would throw off the entire calculation. But we nailed it: $(-2)^2 = 4$. So, the output of our inner function, $g$, when given the input $-2$, is $6$. This value, $6$, now becomes the input for our outer function, $f$. We're one step closer to finding $f(g(-2))$. It's always good practice to double-check this initial calculation. Did we substitute correctly? Did we perform the exponentiation correctly? Yes, we did. $(-2)^2$ is indeed $4$, and adding $2$ gives us $6$. This is the number we'll be working with in the next stage of our problem. Keep this number $6$ handy, as it's the key to unlocking the final answer.

Step 2: Evaluate the Outer Function, $f(6)$

Now that we know $g(-2) = 6$, we need to find the value of $f(6)$. Our function $f(x)$ is given by f(x)=-rac{1}{2} x^2+5 x. We will substitute $6$ for every $x$ in the expression for $f(x)$. So, we get f(6) = -rac{1}{2} (6)^2 + 5(6). Let's break this down. First, we calculate $(6)^2$, which is $6 imes 6 = 36$. Next, we multiply this by -rac{1}{2}. So, -rac{1}{2} imes 36 is the same as dividing $36$ by $2$ and then making it negative, which gives us $-18$. Now for the second term: $5(6)$ is simply $5 imes 6 = 30$. Finally, we add these two results together: $f(6) = -18 + 30$. Adding $-18$ and $30$ gives us $12$. So, the value of $f(g(-2))$ is $12$. It's really satisfying when all the steps come together to give a clean, whole number answer, right? We took the output of $g(-2)$, which was $6$, and plugged it into $f(x)$. The calculation involved squaring $6$ to get $36$, multiplying that by $-\frac{1}{2}$ to get $-18$, and then adding $5$ times $6$, which is $30$. The sum $-18 + 30$ equals $12$. This final result, $12$, is our answer for $f(g(-2))$. Remember to pay attention to the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). In our case, we had exponents first ($6^2$), then multiplications ($-\frac{1}{2} imes 36$ and $5 imes 6$), and finally addition ($-18 + 30$). Following these rules ensures accuracy in our calculations. We have successfully navigated the process of function composition!

Conclusion: The Value of $f(g(-2))$ is 12

So there you have it, folks! We've successfully calculated $f(g(-2))$ by first finding $g(-2)$ and then plugging that result into $f(x)$. The final answer we arrived at is $12$. This problem highlights the importance of understanding function notation and how to perform substitution within functions. It's a fundamental skill in mathematics that opens doors to more complex concepts. Whether you're just starting with algebra or looking to solidify your understanding, practicing these types of problems is key. Remember, the process is: evaluate the inner function first, then use that result as the input for the outer function. Don't be afraid to break down complex problems into smaller, manageable steps. Each step builds upon the previous one, leading you closer to the solution. If you ever get stuck, just retrace your steps, check your arithmetic, and make sure you're applying the function definitions correctly. The beauty of mathematics lies in its logical structure; if you follow the rules, you'll find the answer. Keep practicing, keep exploring, and never hesitate to ask questions. This journey into functions is just the beginning, and there's so much more cool stuff to discover! We've successfully determined that for the given functions $f(x)=-\frac{1}{2} x^2+5 x$ and $g(x)=x^2+2$, the value of $f(g(-2))$ is precisely $12$. It's a clear demonstration of how function composition works: the output of one function becomes the input for another. This concept is incredibly powerful and is used extensively in calculus, physics, computer science, and many other fields. So, give yourself a pat on the back for tackling this! You've reinforced a vital mathematical skill. Keep up the great work, and happy problem-solving!