Finding The Inverse Of F(x) = (2x - 4) / (-3x + 1) A Step-by-Step Guide
Understanding Inverse Functions
In mathematics, particularly in algebra and calculus, the concept of an inverse function is fundamental. An inverse function essentially undoes the operation of the original function. If we apply a function f to a value x and then apply its inverse, denoted as f⁻¹, to the result, we should get back the original value x. This relationship can be expressed mathematically as f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. The process of finding an inverse function involves a series of algebraic manipulations that effectively reverse the roles of the input and output variables.
Why Find Inverse Functions?
Inverse functions are not merely theoretical constructs; they have significant practical applications across various fields. In cryptography, inverse functions are used in encryption and decryption processes. In computer graphics, they help in transforming coordinate systems and manipulating images. In economics, they can be used to model supply and demand relationships. Understanding inverse functions is also crucial for solving equations, particularly in calculus where they are essential for integration techniques.
Core Concepts and Notation
Before diving into the specific example, it's important to clarify some core concepts and notation. The inverse of a function f is typically denoted as f⁻¹. It is crucial to remember that the superscript -1 is a notation for the inverse and does not represent the reciprocal (1/f(x)). The domain of f⁻¹ is the range of f, and the range of f⁻¹ is the domain of f. This is a direct consequence of the inverse function reversing the roles of input and output.
To determine if a function has an inverse, we use the horizontal line test. A function has an inverse if and only if every horizontal line intersects the graph of the function at most once. This is equivalent to saying that the function must be one-to-one (or injective), meaning that no two different inputs produce the same output. If a function fails the horizontal line test, we may be able to restrict its domain to a subset where it does pass the test, thus allowing us to define an inverse on that restricted domain.
Step-by-Step Method to Find the Inverse
Finding the inverse of a function typically involves the following steps:
- Replace f(x) with y: This step simplifies the notation and makes the algebraic manipulations easier to follow.
- Swap x and y: This is the core of the inverse process, as it reverses the roles of input and output.
- Solve for y: This step involves isolating y on one side of the equation, which gives us the expression for the inverse function.
- Replace y with f⁻¹(x): This is the final notational step, expressing the inverse function in standard notation.
Applying the Steps to f(x) = (2x - 4) / (-3x + 1)
Now, let's apply these steps to the given function: f(x) = (2x - 4) / (-3x + 1). This is a rational function, and finding its inverse will involve some algebraic manipulation of fractions.
Step 1: Replace f(x) with y
We begin by replacing f(x) with y:
y = (2x - 4) / (-3x + 1)
This simple substitution makes the equation more visually manageable for the subsequent steps.
Step 2: Swap x and y
Next, we swap x and y:
x = (2y - 4) / (-3y + 1)
This is the key step in finding the inverse, as it reverses the roles of the input and output variables. We are now expressing x in terms of y, which is the foundation for solving for y and finding the inverse function.
Step 3: Solve for y
This is the most algebraically intensive step. We need to isolate y on one side of the equation. First, we multiply both sides by (-3y + 1) to eliminate the denominator:
x(-3y + 1) = 2y - 4
Next, distribute x on the left side:
-3xy + x = 2y - 4
Now, we want to gather all terms containing y on one side of the equation and all other terms on the other side. Add 3xy to both sides and add 4 to both sides:
x + 4 = 2y + 3xy
Factor out y from the right side:
x + 4 = y(2 + 3x)
Finally, divide both sides by (2 + 3x) to isolate y:
y = (x + 4) / (3x + 2)
Step 4: Replace y with f⁻¹(x)
We replace y with f⁻¹(x) to denote the inverse function:
f⁻¹(x) = (x + 4) / (3x + 2)
This is the inverse of the original function f(x) = (2x - 4) / (-3x + 1).
Verifying the Inverse Function
To ensure that we have correctly found the inverse, we can verify our result by checking if f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. Let's perform these compositions.
Checking f⁻¹(f(x))
f⁻¹(f(x)) = f⁻¹((2x - 4) / (-3x + 1))
Substitute (2x - 4) / (-3x + 1) into f⁻¹(x):
f⁻¹(f(x)) = (((2x - 4) / (-3x + 1)) + 4) / (3((2x - 4) / (-3x + 1)) + 2)
To simplify this expression, we need to find a common denominator for the numerator and the denominator. Multiply 4 by (-3x + 1) / (-3x + 1) and 2 by (-3x + 1) / (-3x + 1):
f⁻¹(f(x)) = (((2x - 4) + 4(-3x + 1)) / (-3x + 1)) / ((3(2x - 4) + 2(-3x + 1)) / (-3x + 1))
Simplify the numerator and the denominator:
f⁻¹(f(x)) = ((2x - 4 - 12x + 4) / (-3x + 1)) / ((6x - 12 - 6x + 2) / (-3x + 1))
f⁻¹(f(x)) = ((-10x) / (-3x + 1)) / ((-10) / (-3x + 1))
Divide the fractions by multiplying by the reciprocal of the denominator:
f⁻¹(f(x)) = (-10x) / (-10) = x
So, f⁻¹(f(x)) = x, which is the first condition for verifying the inverse.
Checking f(f⁻¹(x))
Now, let's check the other composition:
f(f⁻¹(x)) = f((x + 4) / (3x + 2))
Substitute (x + 4) / (3x + 2) into f(x):
f(f⁻¹(x)) = (2((x + 4) / (3x + 2)) - 4) / (-3((x + 4) / (3x + 2)) + 1)
To simplify this expression, we need to find a common denominator for the numerator and the denominator. Multiply 4 by (3x + 2) / (3x + 2) and 1 by (3x + 2) / (3x + 2):
f(f⁻¹(x)) = ((2(x + 4) - 4(3x + 2)) / (3x + 2)) / ((-3(x + 4) + (3x + 2)) / (3x + 2))
Simplify the numerator and the denominator:
f(f⁻¹(x)) = ((2x + 8 - 12x - 8) / (3x + 2)) / ((-3x - 12 + 3x + 2) / (3x + 2))
f(f⁻¹(x)) = ((-10x) / (3x + 2)) / ((-10) / (3x + 2))
Divide the fractions by multiplying by the reciprocal of the denominator:
f(f⁻¹(x)) = (-10x) / (-10) = x
So, f(f⁻¹(x)) = x, which is the second condition for verifying the inverse.
Since both f⁻¹(f(x)) = x and f(f⁻¹(x)) = x hold true, we have successfully verified that f⁻¹(x) = (x + 4) / (3x + 2) is indeed the inverse of f(x) = (2x - 4) / (-3x + 1).
Domain and Range of the Inverse Function
To fully understand the inverse function, it's crucial to consider its domain and range. The domain of f⁻¹(x) is all real numbers except for the value that makes the denominator zero. In this case, the denominator 3x + 2 is zero when x = -2/3. Therefore, the domain of f⁻¹(x) is all real numbers except x = -2/3. This is written in interval notation as (-∞, -2/3) ∪ (-2/3, ∞).
The range of f⁻¹(x) is the domain of the original function f(x). The domain of f(x) is all real numbers except for the value that makes its denominator zero. The denominator -3x + 1 is zero when x = 1/3. Therefore, the domain of f(x) is all real numbers except x = 1/3, which is written as (-∞, 1/3) ∪ (1/3, ∞). Thus, the range of f⁻¹(x) is (-∞, 1/3) ∪ (1/3, ∞).
Conclusion
In conclusion, we have successfully found the inverse of the function f(x) = (2x - 4) / (-3x + 1) using the step-by-step method. The inverse function is f⁻¹(x) = (x + 4) / (3x + 2). We verified this result by checking the compositions f⁻¹(f(x)) = x and f(f⁻¹(x)) = x. We also determined the domain and range of the inverse function, which are essential for a complete understanding of the function's behavior. This comprehensive approach provides a solid foundation for working with inverse functions in various mathematical contexts.
Further Exploration
The process of finding inverse functions extends beyond rational functions. It can be applied to polynomial functions, exponential functions, logarithmic functions, and trigonometric functions, among others. Each type of function may require different algebraic techniques to isolate y and find the inverse. For example, finding the inverse of an exponential function often involves using logarithms, while finding the inverse of a trigonometric function requires using inverse trigonometric functions.
Understanding the relationship between a function and its inverse is a cornerstone of mathematical analysis and is crucial for solving a wide range of problems in science, engineering, and other fields. By mastering the techniques for finding inverse functions, you'll be well-equipped to tackle more advanced mathematical concepts and applications.
