Unlocking Functions: Finding G(f(-2)) Step-by-Step

Hey math enthusiasts! Today, we're diving into the world of functions. Specifically, we're going to break down how to find the value of g(f(-2)). Don't worry, it sounds more complicated than it is! This is a core concept in algebra, and understanding it unlocks a whole new level of problem-solving. We'll be using two functions: f(x) and g(x). Our mission? To figure out what happens when we first apply f to -2, and then apply g to the result.

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

Let's start by defining our players. We've got two functions: f(x) = x² - 5x - 7 and g(x) = 5x - 13. Think of a function like a machine. You put something in (an input), and it spits something else out (an output). In our case, the input is 'x', and the functions perform specific operations on 'x' to give us the output. The beauty of functions is that they allow us to describe relationships between variables in a clear and concise way. They are fundamental in mathematics, showing up everywhere from calculus to computer science, making them super important to grasp.

f(x) is a quadratic function, characterized by the x² term. This means its graph is a parabola (a U-shaped curve). The other function, g(x), is a linear function; the graph is a straight line. What makes this problem interesting is that we're going to use one function's output as the input for another function. This concept of function composition is key! It opens doors to more intricate mathematical models and scenarios. Remember, functions aren't just for abstract math; they model real-world phenomena too, from predicting stock prices to understanding population growth. This is the first, crucial step toward mastering functions, allowing you to build a sturdy foundation for more complex mathematical ideas.

Step 1: Evaluating f(-2) - The First Machine

Alright, let's start with f(-2). This means we're going to substitute -2 for 'x' in the f(x) function. Where ever we see 'x' we will substitute it with -2. This is the heart of function evaluation, where we plug specific values into the function to obtain corresponding outputs. It is akin to feeding an ingredient into a recipe, and getting a dish out. We get: f(-2) = (-2)² - 5(-2) - 7*. Let's break this down further.

First, square -2: (-2)² = 4. Then multiply -5 by -2: -5 * -2 = 10. Now, put it all together: f(-2) = 4 + 10 - 7. That is, f(-2) = 7. This is our intermediate result. We've taken the input (-2) and put it through the f machine, and the output is 7. That's the f(x) machine's output. So, essentially, f(-2) simplifies to 7. This step highlights the practical side of functions – they transform inputs into outputs, each step in the process contributing to the final result. Understanding this process solidifies the concept of functions as transformers of values. This transformation is pivotal; it's how we model change and relationships.

Remember, in the vast world of functions, each operation has a specific purpose, contributing to the ultimate transformation of the input. Now, we are ready to take this result and use it as an input into the next function.

Step 2: Evaluating g(f(-2)) - The Second Machine

Now, we're not done yet, because the question asks us to find g(f(-2)). We have just found out that f(-2) = 7. So, we can replace f(-2) with 7. This means we need to find g(7). We are going to substitute 7 for 'x' in the g(x) function. So g(7) = 5(7) - 13*. Let's solve this! We multiply 5 by 7, which gives us 35. Then, subtract 13: 35 - 13 = 22. Therefore, g(7) = 22. Therefore, g(f(-2)) = 22. So, we have finally found our answer! We've successfully composed the functions f and g. We used the output of f as the input to g. Now, we have successfully navigated through the realm of function composition, a critical tool in mathematical problem-solving.

This final step emphasizes how functions work in concert. The output of one function feeds into another, creating complex operations. Mastery of function composition expands the horizons of mathematical manipulation, allowing us to solve intricate problems. This approach allows us to solve intricate problems, providing a clear path to the final solution. The connection between f(x) and g(x) is the core of this operation, highlighting the power of functions in transforming values.

Conclusion: We Did It!

So, there you have it, guys! We have found that g(f(-2)) = 22. By breaking down the problem step-by-step, we've demonstrated how to evaluate composite functions. Function composition is a fundamental skill in algebra, setting you up for success in more complex topics like calculus and beyond. Keep practicing, and you'll become a pro at these problems! Functions are versatile mathematical tools, capable of representing a broad spectrum of real-world scenarios. With practice and persistence, these concepts become second nature.

Mastering function evaluation, like we did with f(-2), and function composition, like with g(f(-2)), are stepping stones to success in higher-level math. Each problem strengthens your problem-solving skills, and reinforces your understanding of the underlying mathematical principles. Keep up the great work, and happy calculating!