Finding (f ∘ G)(-1) Given F(x) And G(x)

Hey guys! Today, we're diving into a fun little problem involving composite functions. Specifically, we're tasked with finding the value of (f ∘ g)(-1) given the definitions of two functions, f(x) and g(x). Don't worry, it sounds fancier than it actually is! We'll break it down step-by-step so it's super easy to follow. So let's jump right into the heart of composite functions and how to solve for them! At first, composite functions might seem a bit intimidating, but they're really just a way of combining two functions into one. Think of it like a mathematical assembly line. One function does its thing, and then the output becomes the input for the next function. This chaining of functions is what makes composite functions so versatile and useful in various areas of mathematics and beyond. They allow us to model complex relationships and processes by breaking them down into smaller, more manageable steps. So, let's get started and demystify the process of evaluating composite functions, one step at a time.

Understanding the Functions

First, let's clearly define the functions we're working with. We have:

• f(x) = -x - 12
• g(x) = x² - 5x - 5

These are our two main players. The function f(x) takes an input x, multiplies it by -1, and then subtracts 12. The function g(x) takes an input x, squares it, subtracts 5 times x, and then subtracts 5 again. Understanding what each function does individually is the crucial first step before we can combine them. Think of it like understanding the individual ingredients before you bake a cake. Each ingredient has its own role, and only when combined in the right way do you get the desired outcome. Similarly, each function has its own operation, and understanding these operations allows us to correctly combine them in a composite function. So let's make sure we've got a solid grasp on what f(x) and g(x) do on their own, before we start mixing them together.

Diving Deeper into f(x)

The function f(x) = -x - 12 is a linear function, characterized by its constant rate of change. This means that for every increase in x, the value of f(x) decreases by a constant amount. The negative sign in front of the x indicates that the function is decreasing, or has a negative slope. The -12 represents the y-intercept, which is the point where the line crosses the y-axis. Understanding linear functions is fundamental in mathematics as they model many real-world phenomena, such as the relationship between distance and time at a constant speed. They are also the building blocks for more complex functions. In the context of our problem, f(x) acts as a transformation that flips the input x across the y-axis (due to the negative sign) and then shifts it down by 12 units. This transformation is important to consider when we compose f(x) with another function, as it will affect the overall behavior of the composite function.

A Closer Look at g(x)

On the other hand, g(x) = x² - 5x - 5 is a quadratic function. This means its graph is a parabola, a U-shaped curve. The x² term is what makes it quadratic, and it's responsible for the curve shape. The other terms, -5x and -5, shift and position the parabola in the coordinate plane. Quadratic functions are used to model various physical phenomena, such as the trajectory of a projectile or the shape of a satellite dish. They also appear frequently in optimization problems, where we want to find the maximum or minimum value of a function. In the context of our problem, g(x) represents a more complex transformation than f(x). It involves squaring the input, which introduces non-linearity, and then subtracting 5 times the input and 5. This means that the output of g(x) will change in a more complex way as the input x changes. Understanding the behavior of g(x) is crucial for correctly evaluating the composite function, as it will determine the input to f(x).

Understanding Composite Functions

Now, let's talk about what (f ∘ g)(-1) actually means. The symbol ∘ represents function composition. This means we're plugging the function g(x) into the function f(x). So, (f ∘ g)(x) is read as "f of g of x," and it's equivalent to f(g(x)). This notation tells us the order in which we apply the functions: first, we evaluate g(x), and then we take that result and plug it into f(x). It's like a two-step process where the output of one function becomes the input of the other. The key to understanding composite functions is to remember the order of operations. We always work from the inside out. So in this case, we start with g(x) and then move to f(x). This is a fundamental concept in mathematics and is used extensively in calculus and other advanced topics. So let's make sure we're clear on the order before we move on to the actual calculation.

Breaking Down (f ∘ g)(-1)

So, (f ∘ g)(-1) means we need to do the following:

1. Find the value of g(-1). This means we substitute -1 for x in the expression for g(x).
2. Take the result from step 1 and plug it into f(x). This means we substitute the value we found in step 1 for x in the expression for f(x).

It's like a recipe: we have two functions, and we're following the instructions to combine them in a specific way. The -1 is our starting ingredient, and we're going to process it through the g function first, and then the f function. This step-by-step approach is crucial for solving composite function problems correctly. We can't skip steps or do them out of order, or we'll end up with the wrong answer. So let's take it slow and make sure we understand each step before moving on. This methodical approach will not only help us solve this problem but also build a solid foundation for understanding more complex mathematical concepts.

Step 1: Evaluating g(-1)

Let's start by finding g(-1). We substitute -1 for x in the expression for g(x):

g(-1) = (-1)² - 5(-1) - 5

Now, we just need to simplify this expression. Remember the order of operations (PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), Addition and Subtraction (from left to right).

g(-1) = 1 + 5 - 5 g(-1) = 1

So, we've found that g(-1) = 1. This is a crucial intermediate result. It's the output of the first function in our composition, and it will become the input for the second function. It's like the first stop on our assembly line. We've taken the raw material (-1) and processed it through the g function, and now we have a partially finished product (1). This value is now ready to be fed into the f function. Making sure we get this value right is essential, as any error here will propagate through the rest of the calculation. So let's double-check our work and make sure we're confident with our result before moving on to the next step.

Step 2: Evaluating f(g(-1))

Now that we know g(-1) = 1, we can plug this value into f(x) to find f(g(-1)). Remember, this is the same as f(1):

f(g(-1)) = f(1) = -(1) - 12

Again, we just need to simplify:

f(1) = -1 - 12 f(1) = -13

And there we have it! We've found that f(g(-1)) = -13. This is the final step in our calculation. We've taken the output of the g function (1) and used it as the input for the f function. This completes the composite function operation. It's like the final stop on our assembly line. We've taken the partially finished product (1) and processed it through the f function, and now we have the final product (-13). This is the value of the composite function at x = -1. So, we can confidently say that the value of (f ∘ g)(-1) is -13.

Final Answer

Therefore, (f ∘ g)(-1) = -13

We did it! By breaking down the problem into smaller steps and carefully following the order of operations, we were able to successfully evaluate the composite function. Remember, the key to composite functions is to work from the inside out. First, evaluate the inner function, and then use its output as the input for the outer function. This process might seem a bit tricky at first, but with practice, it becomes second nature. So keep practicing, and you'll be a composite function master in no time!

So, there you have it, guys! We've successfully navigated the world of composite functions and found the value of (f ∘ g)(-1). I hope this breakdown was helpful and made the concept a little less daunting. Keep practicing, and you'll be composing functions like a pro in no time!