Finding Inverse Function Domain And Range Of F(x) = 2x / (5x-3)

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In mathematics, understanding the concept of inverse functions is crucial, especially when dealing with one-to-one functions. A one-to-one function, also known as an injective function, ensures that each element of the range corresponds to exactly one element of the domain. This property allows us to define an inverse function, denoted as f-1(x), which essentially reverses the operation of the original function f(x). In this article, we will delve into the process of finding the inverse of a given one-to-one function, along with determining its domain and range. Specifically, we will focus on the function f(x) = (2x) / (5x - 3).

Understanding Inverse Functions

Before diving into the specifics, let's solidify our understanding of inverse functions. If f(x) maps an input x to an output y, then the inverse function f-1(x) maps that output y back to the original input x. Mathematically, this can be expressed as follows:

  • If f(x) = y, then f-1(y) = x

This fundamental relationship is key to finding the inverse function. The domain of f(x) becomes the range of f-1(x), and the range of f(x) becomes the domain of f-1(x). This swapping of domain and range is a characteristic feature of inverse functions.

Steps to Find the Inverse Function

  1. Replace f(x) with y: This simplifies the notation and makes the equation easier to manipulate.
  2. Swap x and y: This is the core step in finding the inverse, as it reflects the reversing of the function's operation.
  3. Solve for y: Isolate y on one side of the equation. This will give you the expression for f-1(x).
  4. Replace y with f-1(x): This completes the process of expressing the inverse function.

Finding the Inverse of f(x) = (2x) / (5x - 3)

Now, let's apply these steps to the given function f(x) = (2x) / (5x - 3).

  1. Replace f(x) with y: y = (2x) / (5x - 3)

  2. Swap x and y: x = (2y) / (5y - 3)

  3. Solve for y: To isolate y, we'll first multiply both sides by (5y - 3): x(5y - 3) = 2y Expand the left side: 5xy - 3x = 2y Now, move all terms containing y to one side and all other terms to the other side: 5xy - 2y = 3x Factor out y: y(5x - 2) = 3x Finally, divide both sides by (5x - 2) to solve for y: y = (3x) / (5x - 2)

  4. Replace y with f-1(x):Therefore, the inverse function is:

    f-1(x) = (3x) / (5x - 2)

Determining the Domain and Range of f-1(x)

To determine the domain and range of the inverse function, we need to consider the original function and its inverse.

Domain of f-1(x)

The domain of f-1(x) is the set of all possible input values (x) for which the function is defined. Looking at the expression f-1(x) = (3x) / (5x - 2), we see that the denominator cannot be zero. Therefore, we need to find the value(s) of x that make the denominator zero and exclude them from the domain.

Set the denominator equal to zero and solve for x:

5x - 2 = 0

5x = 2

x = 2/5

Thus, x cannot be equal to 2/5. In interval notation, the domain of f-1(x) is:

Domain of f-1(x) : (-∞, 2/5) ∪ (2/5, ∞)

Range of f-1(x)

The range of f-1(x) is the set of all possible output values (y) that the function can produce. As mentioned earlier, the range of f-1(x) is the same as the domain of the original function f(x). To find the domain of f(x) = (2x) / (5x - 3), we again look for values of x that would make the denominator zero:

5x - 3 = 0

5x = 3

x = 3/5

So, x cannot be equal to 3/5 in the original function. Therefore, the range of f-1(x) is all real numbers except 3/5. In interval notation:

Range of f-1(x) : (-∞, 3/5) ∪ (3/5, ∞)

Summary and Conclusion

In summary, we have successfully found the inverse of the function f(x) = (2x) / (5x - 3) to be f-1(x) = (3x) / (5x - 2). We then determined the domain of f-1(x) to be (-∞, 2/5) ∪ (2/5, ∞) and the range to be (-∞, 3/5) ∪ (3/5, ∞). This process highlights the crucial relationship between a function and its inverse, particularly the swapping of their domains and ranges.

Understanding inverse functions is vital in various areas of mathematics, including calculus, algebra, and analysis. By mastering the techniques for finding inverses and determining their domains and ranges, you can gain a deeper appreciation for the interconnectedness of mathematical concepts and enhance your problem-solving abilities. Remember to always check for restrictions on the domain, especially when dealing with rational functions, as these restrictions directly impact the range of the inverse function. Inverse functions play a significant role in mathematical analysis and are essential for solving complex equations and understanding functional relationships. The ability to find and analyze inverse functions is a fundamental skill in mathematics.

Importance of Understanding Domain and Range

Determining the domain and range of a function, especially an inverse function, is not just a procedural step; it's a crucial aspect of understanding the function's behavior and limitations. The domain tells us the set of valid inputs, while the range tells us the set of possible outputs. Failing to consider the domain can lead to incorrect results or misinterpretations of the function's behavior. For instance, in the function f-1(x) = (3x) / (5x - 2), if we were to plug in x = 2/5 without considering the domain, we would encounter a division by zero, which is undefined. Similarly, understanding the range helps us know what values the function can actually produce.

Real-World Applications

The concept of inverse functions is not limited to theoretical mathematics; it has practical applications in various fields. For example, in cryptography, inverse functions are used to encrypt and decrypt messages. The encryption process applies a function to the original message (plaintext) to transform it into an unreadable form (ciphertext). The decryption process then uses the inverse function to transform the ciphertext back into the original plaintext. In computer graphics, inverse functions are used in transformations, such as rotations and scaling, to map objects back to their original positions or sizes.

In economics, inverse functions can be used to model supply and demand curves. The demand curve represents the quantity of a product that consumers are willing to buy at a given price, while the supply curve represents the quantity that producers are willing to sell at that price. The inverse of the demand function can be used to determine the price at which a certain quantity will be demanded, and the inverse of the supply function can be used to determine the price at which a certain quantity will be supplied. These applications underscore the importance of understanding inverse functions in various real-world scenarios.

Further Exploration

To further enhance your understanding of inverse functions, consider exploring more complex functions and their inverses. This might involve functions with multiple restrictions on their domains or functions that require more advanced algebraic techniques to find their inverses. Additionally, exploring the graphical representation of functions and their inverses can provide valuable insights into their relationship. The graph of an inverse function is a reflection of the original function across the line y = x. This visual representation can help solidify your understanding of the concept of inverse functions.

In conclusion, the process of finding the inverse of a one-to-one function and determining its domain and range is a fundamental skill in mathematics with wide-ranging applications. By understanding the underlying concepts and mastering the techniques involved, you can enhance your problem-solving abilities and gain a deeper appreciation for the interconnectedness of mathematical ideas. Remember, practice and exploration are key to mastering this and other mathematical concepts.

In this comprehensive guide, we've explored the intricacies of finding the inverse of a one-to-one function, specifically focusing on f(x) = (2x) / (5x - 3). We've meticulously walked through the steps to derive the inverse function, f-1(x) = (3x) / (5x - 2), and delved into the crucial process of determining its domain and range. Understanding these concepts is not just about solving mathematical problems; it's about building a strong foundation for more advanced mathematical studies and real-world applications. The domain and range, as we've seen, are not mere technicalities; they are fundamental to the function's definition and behavior. A function without a defined domain and range is like a map without a scale or direction – it lacks the necessary context to be useful.

By mastering the process of finding inverse functions and understanding their domains and ranges, you equip yourself with a powerful toolset for navigating the world of mathematics and beyond. Whether you're solving equations, modeling real-world phenomena, or exploring advanced mathematical concepts, the principles discussed here will serve you well. Remember, mathematics is not just a collection of formulas and procedures; it's a way of thinking, a way of approaching problems with logic and precision. The journey of mathematical exploration is continuous, and each concept you master opens doors to new and exciting possibilities. Embrace the challenge, keep exploring, and let the beauty of mathematics unfold before you.