Finding The Inverse Of G(x) = (3x-8)/(2-x): Domain & Range

by ADMIN 59 views

Hey guys! Today, we're diving into the world of one-to-one functions and their inverses. Specifically, we're going to tackle the function g(x) = (3x - 8) / (2 - x). Our mission? To find its inverse, g⁻¹(x), and then figure out the domain and range for both g(x) and g⁻¹(x). Buckle up, it's going to be a fun ride!

Understanding One-to-One Functions and Inverses

Before we jump into the nitty-gritty, let's quickly recap what one-to-one functions and their inverses are all about. A function is one-to-one (also called injective) if each element of the range corresponds to exactly one element of the domain. In simpler terms, no two different x-values produce the same y-value. This is crucial because only one-to-one functions have inverses.

The inverse function, denoted as g⁻¹(x), essentially "undoes" what the original function g(x) does. If g(a) = b, then g⁻¹(b) = a. Graphically, the graphs of a function and its inverse are reflections of each other across the line y = x. Understanding these foundational concepts of one-to-one functions is essential before diving into the specifics of finding the inverse and determining the domain and range.

When dealing with the function g(x) = (3x - 8) / (2 - x), verifying that it is indeed one-to-one is the first step. This can be done graphically using the horizontal line test or algebraically by confirming that if g(x₁) = g(x₂), then x₁ = x₂. This ensures that the function has a unique inverse. The process of finding an inverse function involves swapping the roles of x and y in the original function and then solving for y. This new function represents the inverse, and its equation is crucial for subsequent analysis of its properties, especially its domain and range. The relationship between a function and its inverse is symmetrical; understanding this symmetry aids in predicting the characteristics of the inverse function once the original function is known. In our case, this symmetry will be particularly useful in simplifying the process of determining the domain and range of the inverse function.

Part a: Finding the Inverse Function

Okay, let's get our hands dirty and find the inverse function, g⁻¹(x). Here's the breakdown:

  1. Replace g(x) with y: This makes the equation easier to work with. So, we have y = (3x - 8) / (2 - x).
  2. Swap x and y: This is the key step in finding the inverse. We get x = (3y - 8) / (2 - y).
  3. Solve for y: This is where the algebra comes in. We need to isolate y on one side of the equation.
    • Multiply both sides by (2 - y): x(2 - y) = 3y - 8
    • Distribute the x: 2x - xy = 3y - 8
    • Get all the y terms on one side and the non-y terms on the other: 2x + 8 = 3y + xy
    • Factor out y: 2x + 8 = y(3 + x)
    • Divide by (3 + x): y = (2x + 8) / (x + 3)
  4. Replace y with g⁻¹(x): We've found our inverse function! g⁻¹(x) = (2x + 8) / (x + 3)

So, the inverse function is g⁻¹(x) = (2x + 8) / (x + 3). Each step in this process is crucial for accurately finding the inverse function. The initial swap of variables, x and y, is based on the principle that the inverse function essentially reverses the roles of the input and output of the original function. The algebraic manipulation required to solve for y can sometimes be tricky, especially when dealing with rational functions. It involves careful application of algebraic rules to ensure that y is properly isolated. Factoring and distributing terms correctly are essential skills in this process. Once the equation is solved for y, replacing y with the notation g⁻¹(x) formally defines the inverse function, making it clear that this new function is the inverse of the original function, g(x).

Part b: Domain and Range

Now, let's talk domain and range. Remember, the domain is the set of all possible input values (x-values), and the range is the set of all possible output values (y-values).

For g(x) = (3x - 8) / (2 - x)

  • Domain: The only thing that could stop us here is division by zero. So, we need to find the x-value that makes the denominator (2 - x) equal to zero. 2 - x = 0 implies x = 2. Therefore, the domain of g(x) is all real numbers except 2. We can write this as (-∞, 2) ∪ (2, ∞).
  • Range: To find the range, it's helpful to consider the horizontal asymptote of the function. As x gets very large (positive or negative), the function approaches the ratio of the leading coefficients, which is 3 / -1 = -3. This means y will never actually equal -3. So, the range of g(x) is all real numbers except -3. We can write this as (-∞, -3) ∪ (-3, ∞). Determining the domain of g(x) involves identifying values of x that would lead to undefined operations, such as division by zero. Setting the denominator equal to zero and solving for x reveals these problematic values. The range, on the other hand, requires a deeper understanding of the function's behavior, particularly its asymptotic behavior. Horizontal asymptotes dictate the values that the function approaches as x tends to infinity or negative infinity. Analyzing these asymptotes is crucial for identifying the bounds of the range. For rational functions, the ratio of the leading coefficients of the numerator and the denominator often provides a quick way to find the horizontal asymptote.

For g⁻¹(x) = (2x + 8) / (x + 3)

Here's where a cool trick comes in handy: 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 is a fundamental property of inverse functions!

  • Domain: Since the range of g(x) is (-∞, -3) ∪ (-3, ∞), the domain of g⁻¹(x) is also (-∞, -3) ∪ (-3, ∞). We can also confirm this by looking at the denominator of g⁻¹(x): x + 3 = 0 implies x = -3, so x cannot be -3.
  • Range: Since the domain of g(x) is (-∞, 2) ∪ (2, ∞), the range of g⁻¹(x) is also (-∞, 2) ∪ (2, ∞). Similarly, we can find the horizontal asymptote of g⁻¹(x), which is 2 / 1 = 2, confirming that y will never equal 2. The inverse relationship between the domain and range of a function and its inverse is a powerful concept. It simplifies the process of finding the domain and range of the inverse function once the domain and range of the original function are known. This symmetry is a direct consequence of the swapping of x and y in the process of finding the inverse. However, it is still beneficial to verify these results by analyzing the inverse function directly, especially in more complex cases. For the inverse function g⁻¹(x), determining the values of x that would lead to division by zero confirms the domain, while analyzing its horizontal asymptote confirms the range.

Wrapping Up

So, there you have it! We found the inverse of g(x) = (3x - 8) / (2 - x) to be g⁻¹(x) = (2x + 8) / (x + 3). We also determined that:

  • Domain of g(x): (-∞, 2) ∪ (2, ∞)
  • Range of g(x): (-∞, -3) ∪ (-3, ∞)
  • Domain of g⁻¹(x): (-∞, -3) ∪ (-3, ∞)
  • Range of g⁻¹(x): (-∞, 2) ∪ (2, ∞)

Understanding how to find inverse functions and their domains and ranges is a fundamental skill in mathematics. Keep practicing, and you'll become a pro in no time! Remember the key takeaways: swap x and y to find the inverse, and remember the domain of the original is the range of the inverse (and vice versa). Keep these in mind, and you'll be solving these problems like a champ. The ability to accurately determine the inverse of a function and its corresponding domain and range demonstrates a comprehensive understanding of the fundamental properties of functions. Mastering these skills not only aids in solving mathematical problems but also provides a strong foundation for more advanced topics in calculus and analysis. The process involves algebraic manipulation, understanding asymptotic behavior, and applying the inverse relationship between the domain and range. Regular practice and conceptual understanding are key to mastering these concepts. This knowledge extends beyond pure mathematics and finds applications in various fields, including physics, engineering, and computer science.