Exploring Functions F(x) And G(x) And Their Composition H(x)

In the fascinating world of mathematics, functions play a pivotal role in modeling real-world phenomena and exploring relationships between variables. This article delves into the intricacies of two functions, f(x) and g(x), defined within the realm of real numbers greater than or equal to zero. We will embark on a journey to understand their behavior, analyze their properties, and explore the composition of these functions, leading us to a new function, H(x). This exploration will not only solidify our understanding of functions but also highlight the power of mathematical manipulation and analysis in uncovering hidden relationships.

The main focus of our exploration will be the function f(x) = 3/(x+3), where x is a real number greater than or equal to zero. This seemingly simple rational function holds a wealth of information waiting to be uncovered. We will also examine the function g(x) = 1 + (3/(x+3)), which builds upon the structure of f(x). The composition of these two functions, denoted as H(x) = g(f(x)), will be a key area of investigation, revealing how the output of one function becomes the input of another, creating a chain of mathematical operations. By carefully analyzing these functions and their composition, we aim to gain a deeper appreciation for the elegance and interconnectedness of mathematical concepts.

To truly grasp the essence of these functions, we will employ a variety of mathematical techniques, including algebraic manipulation, function composition, and domain analysis. We will carefully examine the domain of each function, ensuring that our calculations are valid and meaningful. The composition H(x) will be derived step-by-step, showcasing the power of symbolic manipulation in revealing the underlying structure of complex functions. Furthermore, we will explore the implications of the condition x ≥ 0, which restricts the domain of our functions and influences their behavior. This restriction adds a layer of complexity to our analysis, forcing us to consider the specific context in which these functions are defined. The ultimate goal is to provide a comprehensive and insightful exploration of these functions, making the concepts accessible to a broad audience and fostering a deeper appreciation for the beauty of mathematics.

Defining the Functions

Let's begin by formally defining the functions that will be the focus of our analysis. We are given two functions:

• f(x) = 3/(x+3), where x ∈ ℝ and x ≥ 0
• g(x) = 1 + (3/(x+3)), where x ∈ ℝ and x ≥ 0

These functions are defined for all real numbers x greater than or equal to zero. This restriction on the domain is crucial and will influence the behavior of the functions. The function f(x) is a rational function, where a constant (3) is divided by a linear expression (x+3). The function g(x) is a transformation of f(x), where we add 1 to the expression. Understanding the impact of this transformation is a key part of our exploration. We will carefully consider how the domains of these functions restrict their behavior and influence their overall properties. The condition x ≥ 0 implies that we are only concerned with non-negative real numbers, which simplifies our analysis in some ways but also requires us to be mindful of the boundaries of our domain. The behavior of these functions within this specific domain is what we aim to fully understand.

It's important to note that the denominator of both f(x) and g(x) is x+3. Since x ≥ 0, the denominator will always be greater than or equal to 3, ensuring that we never encounter division by zero. This is a critical observation, as division by zero is undefined in mathematics and would invalidate our calculations. The fact that the denominator is always positive within our domain has implications for the range of the functions, as it means that the output of f(x) will always be positive. Similarly, the output of g(x) will always be greater than 1. These insights into the behavior of the functions within their defined domain are essential for a comprehensive understanding.

Deriving H(x) = g(f(x))

Now, let's delve into the composition of these functions. We are interested in finding the function H(x), which is defined as g(f(x)). This means that we will take the output of the function f(x) and use it as the input for the function g(x). This process of function composition is a fundamental concept in mathematics and allows us to create new functions from existing ones. The key idea is to substitute the expression for f(x) into the argument of g(x). This may seem complicated at first, but by carefully following the steps, we can arrive at a simplified expression for H(x).

To find H(x), we substitute f(x) = 3/(x+3) into g(x) = 1 + (3/(x+3)). This gives us:

H(x) = g(f(x)) = g(3/(x+3)) = 1 + (3 / ((3/(x+3)) + 3))

This expression looks complex, but we can simplify it by focusing on the denominator of the second term. We have a fraction within a fraction, which can be simplified by finding a common denominator and combining the terms. The goal is to eliminate the nested fraction and express H(x) in a more manageable form.

Let's simplify the denominator: (3/(x+3)) + 3 = (3 + 3(x+3)) / (x+3) = (3 + 3x + 9) / (x+3) = (3x + 12) / (x+3). Now we can rewrite H(x) as:

H(x) = 1 + (3 / ((3x + 12) / (x+3))) = 1 + (3 * ((x+3) / (3x + 12))) = 1 + ((3(x+3)) / (3(x + 4)))

Notice that we can cancel out the factor of 3 in the numerator and denominator of the second term. This further simplifies our expression:

H(x) = 1 + ((x+3) / (x + 4))

Finally, we can combine the two terms by finding a common denominator:

H(x) = (x + 4 + x + 3) / (x + 4) = (2x + 7) / (x + 4)

Therefore, the composition of the functions f(x) and g(x), H(x) = g(f(x)), is given by H(x) = (2x + 7) / (x + 4). This is a rational function, similar to f(x), but with a different structure. We have successfully derived the expression for H(x) through careful algebraic manipulation and simplification. This result is a significant step in understanding the relationship between f(x) and g(x).

Rewriting H(x) to the desired form

We have found that H(x) = (2x + 7) / (x + 4). The goal is to show that H(x) can be expressed in the form (x+6) / (x+3). To achieve this, we need to manipulate the expression for H(x) algebraically. The key idea is to try and rewrite the numerator in terms of the denominator or vice versa. This often involves adding and subtracting terms strategically to create matching expressions.

Let's start by trying to rewrite the numerator (2x + 7) in terms of (x + 4). We can do this by multiplying (x + 4) by 2, which gives us 2x + 8. Notice that this is very close to our numerator, 2x + 7. We can obtain our numerator by subtracting 1 from 2x + 8. Therefore, we can rewrite the numerator as 2(x + 4) - 1. This allows us to rewrite H(x) as:

H(x) = (2(x + 4) - 1) / (x + 4) = (2(x + 4) / (x + 4)) - (1 / (x + 4)) = 2 - (1 / (x + 4))

While this form is simplified, it doesn't directly match our desired form of (x+6) / (x+3). So, let's try a different approach. We'll go back to the original expression for H(x), (2x + 7) / (x + 4), and try to manipulate it to get the desired form. The strategy this time is to focus on getting an (x+3) term in the denominator.

To do this, we can add and subtract 1 in the denominator: (x + 4) = (x + 3 + 1). This doesn't directly help us simplify the expression, but it keeps in mind the form we want to achieve. Now, let's try to manipulate the numerator to get a term involving (x+3). We can rewrite the numerator as follows:

2x + 7 = 2(x + 3) + 1

This allows us to rewrite H(x) as:

H(x) = (2(x + 3) + 1) / (x + 4)

This form is closer to our goal, but we still need to manipulate it further. We can't directly cancel any terms in this form. Let's try a different approach altogether. Since we want to show that H(x) = (x+6) / (x+3), let's start with the desired form and work backwards. This method can sometimes be helpful in proving identities.

If H(x) = (x+6) / (x+3), then we need to show that (2x + 7) / (x + 4) = (x+6) / (x+3). To do this, we can cross-multiply and see if we get the same expression on both sides:

(2x + 7)(x + 3) = (x + 6)(x + 4)

Expanding both sides, we get:

2x² + 6x + 7x + 21 = x² + 4x + 6x + 24

2x² + 13x + 21 = x² + 10x + 24

These expressions are not equal, which means that our initial hypothesis that H(x) = (x+6) / (x+3) is incorrect. We made an error in our initial derivation of H(x). Let's go back to the step where we simplified H(x):

H(x) = 1 + ((x+3) / (x + 4))

We then combined the terms by finding a common denominator:

H(x) = (x + 4 + x + 3) / (x + 4) = (2x + 7) / (x + 4)

This step is correct. The error must have occurred in the very first calculation where we composed the functions. Let's re-evaluate H(x) = g(f(x)) = g(3/(x+3)) = 1 + (3 / ((3/(x+3)) + 3)).

The error was in the earlier simplification, let's do it carefully now:

H(x) = 1 + (3 / ((3/(x+3)) + 1)) H(x) = 1 + (3 / ((3+x+3)/(x+3)) H(x) = 1 + (3 / ((x+6)/(x+3)) H(x) = 1 + (3(x+3) / (x+6)) H(x) = ((x+6) + 3(x+3)) / (x+6) H(x) = (x+6+3x+9) / (x+6) H(x) = (4x+15) / (x+6)

It seems there was a mistake in the original question and H(x) is not equal to (x+6)/(x+3). Let's proceed to find g^{-1} (x)

Finding the Inverse Function g⁻¹(x)

To find the inverse function, g⁻¹(x), we need to reverse the roles of x and y in the equation y = g(x) and then solve for y. This process involves algebraic manipulation to isolate y on one side of the equation. The resulting expression will represent the inverse function, g⁻¹(x). Finding the inverse function is a fundamental concept in mathematics and allows us to