Unveiling (g ∘ F)(x): A Step-by-Step Mathematical Exploration

Hey math enthusiasts! Let's dive headfirst into a cool problem involving function composition. We're given three functions: $f(x)$, $g(x)$, and $h(x)$. Our main goal is to find the value of $(g \circ f)(x)$ when $x = -3$. This might sound intimidating at first, but trust me, it's like a fun puzzle that we can totally crack together. So, grab your pencils, and let's get started. We'll break down the process step-by-step to make sure everything is crystal clear. This is where the real fun begins! We'll be using our knowledge of functions and their interactions. This concept is a fundamental one in mathematics, and understanding it will give you a significant edge in tackling more complex problems down the road. The beauty of this topic is its simplicity once you grasp the underlying principles. We're going to use the power of substitution, which is super helpful, to find the solution. Let's start with the basics, we'll review the functions, then get right into the process of solving it. I will keep you engaged and entertained during this journey. So buckle up, because we're about to embark on a mathematical adventure.

Understanding the Given Functions: The Foundation

Before we start, let's get acquainted with the functions. We have:

• $f(x) = x^3 - 8x$
• $g(x) = \sqrt{2x}$
• $h(x) = 4x + 7$

Notice that we're only interested in functions $f(x)$ and $g(x)$ for this problem. The function $h(x)$ is there, but it's like a guest who's not invited to the party. The function $f(x)$ is a cubic function, meaning it has a term with $x$ raised to the power of 3. Cubic functions are known for their characteristic 'S' shape when graphed. Function $g(x)$ is a square root function. It only takes non-negative numbers as inputs because you can't take the square root of a negative number (at least, not in the real number system). Understanding these basic characteristics will help us navigate through the problem. This is a crucial step! It is a bit like knowing the ingredients before starting to cook a meal. We need to know what we're working with before we can create the final dish. The more familiar we are with each function, the easier it will be to compose them and evaluate them at a given point. We're not just dealing with abstract symbols here; each function has a specific role and behavior. Keep in mind that the domain of $g(x)$ is all non-negative real numbers, and this will be an important consideration later on. We're going to make sure that the inputs we feed into $g(x)$ are valid so that we don't end up with an undefined result. It is not just about the math; it is about grasping the logic and the reasoning behind it.

Breaking Down the Function Composition: $(g \circ f)(x)$

Now, let's talk about $(g \circ f)(x)$. This notation means that we're going to apply the function $f(x)$ first, and then apply the function $g(x)$ to the result. In simpler terms, we're going to put $f(x)$ inside $g(x)$. Mathematically, this is written as $g(f(x))$. Think of it like a two-step process: first, calculate the output of $f(x)$ for a given value of $x$, and then use that output as the input for $g(x)$. Function composition is a fundamental concept in mathematics. It allows us to combine functions and create new ones. It also lets us analyze complex relationships by breaking them down into simpler steps. When we talk about $(g \circ f)(x)$, we're essentially asking what happens when you perform $f(x)$ and then $g(x)$. The order is critical here. It's like putting on your socks before your shoes. $(g \circ f)(x)$ is generally not the same as $(f \circ g)(x)$, so keep the order in mind. If we had $(f \circ g)(x)$, we'd first apply $g(x)$ and then apply $f(x)$. Understanding the notation and what it signifies is half the battle. This is the heart of function composition. The notation $(g \circ f)(x)$ might seem a little intimidating, but it is just a concise way of saying 'apply $f$ first, then $g$'.

Step-by-Step Evaluation: Finding $(g \circ f)(-3)$

Let's get down to the actual calculation. We want to find $(g \circ f)(-3)$. This means we need to substitute $x = -3$ into $f(x)$ first and then take that result and substitute it into $g(x)$. Here's how we do it:

1. Find $f(-3)$: Substitute $x = -3$ into the equation for $f(x)$.

   $f(-3) = (-3)^3 - 8(-3)$

   $f(-3) = -27 + 24$

   $f(-3) = -3$

2. Find $g(f(-3))$: Now that we know $f(-3) = -3$, we substitute this value into the equation for $g(x)$.

   $g(f(-3)) = g(-3)$

   $g(-3) = \sqrt{2(-3)}$

   $g(-3) = \sqrt{-6}$

Oops! We hit a snag. The square root of a negative number is undefined in the real number system. This means that $(g \circ f)(-3)$ is undefined.

Final Answer

Therefore, $(g \circ f)(-3)$ is Undefined.

Deep Dive into the Solution: Unpacking the Logic

Let's unpack the logic and reasoning behind this solution. Function composition is like a well-oiled machine. It takes an input, processes it through a series of steps, and delivers an output. Understanding how each part of the machine works is the key to mastering this concept. We started by evaluating $f(x)$ at $x = -3$. This gave us $f(-3) = -3$. We then used this output as the input for $g(x)$. However, when we tried to evaluate $g(-3)$, we ran into a problem. Because we cannot take the square root of a negative number in the real number system, the expression $\sqrt{-6}$ is undefined. This is a crucial point, and it highlights the importance of understanding the domains of functions. The domain of $g(x)$ is all non-negative real numbers. If we input a negative number into $g(x)$, the function becomes undefined. It is a bit like trying to fit a square peg into a round hole. The inputs to a function must adhere to the rules that govern the function's operation. We saw that $f(-3)$ produced an output of $-3$, which is not in the domain of $g(x)$. This caused the function to become undefined. So, the final answer isn't just a number; it is a statement about the nature of these functions and their behavior when combined. The answer