Domain Of Inverse Composite Function (f ∘ G)^-1(x)

Hey guys! Let's dive into a fun math problem today involving composite functions and their inverses. We're going to figure out the domain of an inverse composite function. It might sound a little intimidating, but trust me, we'll break it down step by step so it's super easy to understand. So, grab your pencils, and 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 composite functions. Think of it like this: you have two machines, f(x) and g(x). When you compose them, you're essentially feeding the output of one machine into the input of the other. The notation (f ∘ g)(x) means you first apply the function g to x, and then you apply the function f to the result. In simpler terms, it's f(g(x)). Understanding this composition is super crucial for tackling the problem at hand. We'll use this concept extensively as we navigate through the solution, ensuring that we grasp every nuance of the function composition process. Remember, the order matters! (f ∘ g)(x) is generally not the same as (g ∘ f)(x), so keep an eye on which function is applied first. This careful attention to detail will help us avoid common pitfalls and arrive at the correct answer. The concept of composite functions is not just a mathematical abstraction; it has real-world applications in various fields, such as computer science, engineering, and physics. So, mastering this concept will not only help you solve math problems but also broaden your understanding of how different systems interact with each other.

Problem Setup

Okay, let's look at the specific problem we have. We're given two functions:

• f(x) = 6/(x+3), where x ≠ -3
• g(x) = x^2, where x ≥ 0

And our mission, should we choose to accept it, is to find the domain of (f ∘ g)^-1(x). That's the inverse of the composite function. Sounds like a puzzle, right? But don't worry, we'll solve it together. First off, let's focus on the restrictions mentioned for each function. For f(x), the condition x ≠ -3 is crucial because it prevents division by zero, a big no-no in math. For g(x), the restriction x ≥ 0 limits the function to non-negative values, which will play a significant role when we consider the inverse function. The domain of a function is the set of all possible input values (x-values) for which the function is defined. When dealing with composite functions and their inverses, it's essential to keep track of these domains and ranges to ensure that our calculations are valid and our final answer is accurate. By carefully considering the restrictions on the original functions, we can avoid common errors and gain a deeper understanding of the behavior of the composite function and its inverse.

Finding the Composite Function (f ∘ g)(x)

The first step in solving this problem is to find the composite function (f ∘ g)(x). Remember, this means we need to plug g(x) into f(x). So, let's do it:

(f ∘ g)(x) = f(g(x)) = f(x^2) = 6/(x^2 + 3)

Alright, we've got our composite function! Notice that we simply replaced the 'x' in f(x) with g(x), which is x^2. This substitution is the heart of function composition. Isn't it cool how functions can be combined like that? Now, let's think about the domain of this new function. The denominator is x^2 + 3. Since x^2 is always non-negative (for real numbers) and we're adding 3 to it, the denominator will never be zero. That's great news! It means there are no additional restrictions on the domain of (f ∘ g)(x) beyond those already imposed by g(x). The domain of g(x) is x ≥ 0, so that restriction still applies to the composite function. Therefore, the domain of (f ∘ g)(x) is x ≥ 0. This careful consideration of the domain is crucial because it ensures that we're working with valid input values throughout the problem. Understanding the domain of the composite function is not only important for finding its inverse but also for analyzing its behavior and properties, such as its range, intercepts, and asymptotes. By mastering the concept of function composition and its domain, we can tackle more complex mathematical problems with confidence and precision.

Finding the Inverse of the Composite Function (f ∘ g)^-1(x)

Now comes the fun part – finding the inverse of the composite function! To find the inverse, we'll switch x and y in the equation y = 6/(x^2 + 3) and then solve for y. Ready? Here we go:

1. Start with y = 6/(x^2 + 3)
2. Switch x and y: x = 6/(y^2 + 3)
3. Solve for y:
   • Multiply both sides by (y^2 + 3): x(y^2 + 3) = 6
   • Divide both sides by x: y^2 + 3 = 6/x
   • Subtract 3 from both sides: y^2 = 6/x - 3
   • Find a common denominator: y^2 = (6 - 3x)/x
   • Take the square root of both sides: y = ±√((6 - 3x)/x)

So, we have y = ±√((6 - 3x)/x). But remember, the range of the original composite function (f ∘ g)(x) becomes the domain of its inverse. The range of (f ∘ g)(x) is (0, 2], because as x increases from 0, the value of 6/(x^2 + 3) decreases from 2 towards 0. Since g(x) = x^2 and x ≥ 0, g(x) will always produce non-negative values. Therefore, when we plug g(x) into f(x), the result will always be positive. This means we only consider the positive square root for the inverse function. Why? Because the original function g(x) had the restriction x ≥ 0, which means the inverse function will also have a range that is non-negative. Thus, the inverse function is:

(f ∘ g)^-1(x) = √((6 - 3x)/x)

Finding the inverse of a function is like reversing a process. We're essentially undoing what the original function did. This concept is widely used in mathematics and computer science to solve equations, decrypt codes, and perform various other tasks. The key to finding the inverse is to switch the roles of the input and output variables (x and y) and then solve for the new output variable (y). This process can sometimes be tricky, especially when dealing with more complex functions, but with practice and a solid understanding of the underlying principles, it becomes second nature.

Determining the Domain of (f ∘ g)^-1(x)

Okay, we're in the home stretch! Now we need to find the domain of (f ∘ g)^-1(x) = √((6 - 3x)/x). This is where things get a little tricky, but we can do it! For the square root to be defined, the expression inside the square root must be non-negative, meaning (6 - 3x)/x ≥ 0. Also, the denominator cannot be zero, so x ≠ 0.

Let's analyze the inequality (6 - 3x)/x ≥ 0:

We need to consider the sign of both the numerator (6 - 3x) and the denominator (x). The numerator is zero when 6 - 3x = 0, which means x = 2. The denominator is zero when x = 0.

Now, let's create a sign chart:

From the sign chart, we can see that (6 - 3x)/x ≥ 0 when 0 < x ≤ 2. We include x = 2 because the expression is equal to zero there, but we exclude x = 0 because it makes the denominator zero. So, the domain of (f ∘ g)^-1(x) is 0 < x ≤ 2. Woohoo, we found it!

Determining the domain of a function is a fundamental step in mathematical analysis. It ensures that we're working with valid input values and that our results are meaningful. When dealing with functions involving square roots, fractions, or logarithms, it's crucial to identify any restrictions on the input values that would lead to undefined or imaginary results. Sign charts, like the one we used here, are a powerful tool for analyzing inequalities and determining the intervals where a function is positive, negative, or zero. By mastering these techniques, we can confidently tackle a wide range of problems involving function domains and inequalities.

Final Answer

So, after all that awesome math work, we've found that the domain of the inverse composite function (f ∘ g)^-1(x) is 0 < x ≤ 2. Give yourselves a pat on the back! You've successfully navigated the world of composite functions and their inverses. Remember, the key to solving these types of problems is to break them down into smaller, more manageable steps. Start by understanding the definitions of the functions involved, then carefully apply the rules of composition and inversion. And most importantly, don't forget to consider the domains and ranges of the functions at each step. By following these guidelines, you'll be well-equipped to tackle even the most challenging math problems. Keep up the great work, guys! And remember, math is not just about finding the right answer; it's about developing critical thinking skills and problem-solving strategies that can be applied to various aspects of life. So, embrace the challenge, enjoy the process, and never stop exploring the fascinating world of mathematics.