Finding The Inverse Of G(x) = (3x - 5) / (x + 6) Domain And Range

In the realm of mathematics, functions play a pivotal role in describing relationships between variables. Among these, one-to-one functions hold a special significance due to their invertibility. This article delves into the intricacies of a specific one-to-one function, g(x) = (3x - 5) / (x + 6), with the aim of finding its inverse, g^-1(x), and exploring the domain and range of this inverse function. Understanding these concepts is crucial for a solid foundation in mathematical analysis and its applications.

Unveiling the Essence of One-to-One Functions and Inverses

Before we embark on the journey of finding the inverse of our given function, it's essential to grasp the fundamental concepts of one-to-one functions and their inverses. A function is deemed one-to-one (also known as injective) if each element in its range corresponds to a unique element in its domain. In simpler terms, no two distinct inputs produce the same output. This property is crucial because it ensures the existence of an inverse function.

An inverse function, denoted as g^-1(x), essentially reverses the operation of the original function, g(x). If g(a) = b, then g^-1(b) = a. This inverse relationship allows us to "undo" the effect of the original function, providing a powerful tool for solving equations and analyzing mathematical models. The domain of a function is the set of all possible input values (x-values), while the range is the set of all possible output values (y-values). When we find the inverse of a function, the domain and range swap roles: the domain of g(x) becomes the range of g^-1(x), and vice versa. This reciprocal relationship is a key aspect of understanding inverse functions.

To determine if a function is one-to-one, we can employ the horizontal line test. If any horizontal line intersects the graph of the function at more than one point, then the function is not one-to-one. Alternatively, we can use an algebraic approach. If we assume g(x₁) = g(x₂) and can prove that x₁ = x₂, then the function is one-to-one. This algebraic method provides a rigorous way to confirm the one-to-one nature of a function.

The process of finding an inverse function involves several key steps. First, we replace g(x) with y. Then, we swap x and y. Next, we solve the resulting equation for y. Finally, we replace y with g^-1(x). These steps systematically lead us to the inverse function, allowing us to reverse the operation of the original function. The domain and range of the original function and its inverse are intricately linked, providing valuable insights into the behavior of both functions.

Finding the Inverse Function g^-1(x) for g(x) = (3x - 5) / (x + 6)

Now, let's apply these concepts to our specific function, g(x) = (3x - 5) / (x + 6). Our goal is to find its inverse, g^-1(x). To achieve this, we'll follow the steps outlined earlier:

1. Replace g(x) with y:

   • We begin by rewriting the function as y = (3x - 5) / (x + 6).

2. Swap x and y:

   • Next, we interchange the variables x and y, resulting in the equation x = (3y - 5) / (y + 6).

3. Solve for y:

   • This is the crucial step where we isolate y. To do this, we first multiply both sides of the equation by (y + 6):
     • x(y + 6) = 3y - 5
   • Expanding the left side gives us:
     • xy + 6x = 3y - 5
   • Now, we gather all terms containing y on one side and the remaining terms on the other side:
     • xy - 3y = -6x - 5
   • We factor out y from the left side:
     • y(x - 3) = -6x - 5
   • Finally, we divide both sides by (x - 3) to solve for y:
     • y = (-6x - 5) / (x - 3)

4. Replace y with g^-1(x):

   • We replace y with g^-1(x) to denote the inverse function:
     • g^-1(x) = (-6x - 5) / (x - 3)

Therefore, the inverse function of g(x) = (3x - 5) / (x + 6) is g^-1(x) = (-6x - 5) / (x - 3). This inverse function allows us to reverse the operation of the original function, providing a powerful tool for solving equations and analyzing mathematical models.

Determining the Domain and Range of g^-1(x)

Having found the inverse function, g^-1(x) = (-6x - 5) / (x - 3), our next task is to determine its domain and range. As mentioned earlier, the domain of g^-1(x) is the range of g(x), and the range of g^-1(x) is the domain of g(x). This reciprocal relationship simplifies our task.

Finding the Domain of g^-1(x)

The domain of g^-1(x) consists of all real numbers except those that make the denominator equal to zero. In our case, the denominator is (x - 3). Setting this equal to zero, we find x = 3. Therefore, the domain of g^-1(x) is all real numbers except 3. In interval notation, this is expressed as (-∞, 3) ∪ (3, ∞). This means that g^-1(x) is defined for all real numbers less than 3 and all real numbers greater than 3, but not at 3 itself.

Finding the Range of g^-1(x)

The range of g^-1(x) is equal to the domain of g(x). To find the domain of g(x) = (3x - 5) / (x + 6), we again look for values that make the denominator zero. The denominator is (x + 6), which equals zero when x = -6. Therefore, the domain of g(x) is all real numbers except -6. This means the range of g^-1(x) is also all real numbers except -6. In interval notation, this is expressed as (-∞, -6) ∪ (-6, ∞). This indicates that the output values of g^-1(x) can be any real number less than -6 or greater than -6, but not -6 itself.

In summary, the domain of g^-1(x) = (-6x - 5) / (x - 3) is (-∞, 3) ∪ (3, ∞), and the range is (-∞, -6) ∪ (-6, ∞). These intervals define the set of permissible input and output values for the inverse function, providing a comprehensive understanding of its behavior.

Conclusion: The Significance of Inverse Functions

In this article, we have successfully navigated the process of finding the inverse of the one-to-one function g(x) = (3x - 5) / (x + 6). We determined that the inverse function is g^-1(x) = (-6x - 5) / (x - 3), and we meticulously identified its domain as (-∞, 3) ∪ (3, ∞) and its range as (-∞, -6) ∪ (-6, ∞). This exercise highlights the crucial relationship between a function and its inverse, demonstrating how they essentially "undo" each other's operations.

The concept of inverse functions is not merely a theoretical curiosity; it has profound implications in various fields of mathematics and its applications. Inverse functions are essential for solving equations, particularly those involving complex expressions. They also play a vital role in calculus, where they are used to find derivatives and integrals of certain functions. Furthermore, inverse functions are crucial in cryptography, where they are used to encrypt and decrypt messages.

By understanding the properties and behavior of inverse functions, we gain a deeper appreciation for the interconnectedness of mathematical concepts. The ability to find and analyze inverse functions empowers us to solve a wider range of problems and to model real-world phenomena more effectively. As we continue our mathematical journey, the knowledge and skills acquired in this exploration will undoubtedly serve as a valuable foundation for future endeavors.