Finding The Inverse: F(x) = (2x - 10)³ Explained

by Alex Johnson 49 views

In mathematics, finding the inverse of a function is a fundamental concept. The inverse function, denoted as f⁻¹(x), essentially undoes what the original function f(x) does. This article will guide you through the process of finding the inverse of the function f(x) = (2x - 10)³, providing a step-by-step explanation and highlighting key concepts along the way. Understanding inverse functions is crucial for various mathematical applications, including solving equations and analyzing relationships between variables.

Understanding Inverse Functions

Before diving into the specific example, let's first understand what an inverse function truly represents. An inverse function reverses the operation of the original function. If f(a) = b, then f⁻¹(b) = a. Think of it as a two-way street: f takes you from 'a' to 'b', and f⁻¹ takes you back from 'b' to 'a'. Graphically, the inverse function is a reflection of the original function across the line y = x. This means that if you were to fold the graph along the line y = x, the original function and its inverse would perfectly overlap. A function must be one-to-one (meaning it passes both the vertical and horizontal line tests) to have a true inverse. This is because each input must correspond to a unique output, and vice versa, for the inverse to be well-defined. If a function isn't one-to-one, we might need to restrict its domain to find a partial inverse.

Steps to Find the Inverse Function

Now, let's break down the process of finding the inverse function into clear, manageable steps. We'll use the given function, f(x) = (2x - 10)³, as our example.

  1. Replace f(x) with y: This is a simple notational change to make the equation easier to work with. So, we rewrite f(x) = (2x - 10)³ as y = (2x - 10)³.

  2. Swap x and y: This is the crucial step in finding the inverse. We're essentially reversing the roles of input and output. By swapping x and y, we get x = (2y - 10)³.

  3. Solve for y: Now, we need to isolate y in the equation. This involves performing algebraic operations to get y by itself on one side of the equation. This is where our algebraic skills come into play, and we'll need to carefully undo the operations applied to y.

  4. Replace y with f⁻¹(x): Once we've solved for y, we replace it with the notation f⁻¹(x) to indicate that we've found the inverse function. This is the final step in expressing the inverse function in standard notation. Remember, f⁻¹(x) represents the inverse function, not 1 divided by f(x).

Applying the Steps to f(x) = (2x - 10)³

Let's apply these steps to find the inverse of our given function, f(x) = (2x - 10)³.

  1. Replace f(x) with y: y = (2x - 10)³

  2. Swap x and y: x = (2y - 10)³

  3. Solve for y:

    • To isolate y, we first need to get rid of the cube. We do this by taking the cube root of both sides of the equation: ∛x = ∛((2y - 10)³) ∛x = 2y - 10
    • Next, we add 10 to both sides: ∛x + 10 = 2y
    • Finally, we divide both sides by 2 to solve for y: y = (∛x + 10) / 2

  4. Replace y with f⁻¹(x): f⁻¹(x) = (∛x + 10) / 2

Therefore, the inverse of the function f(x) = (2x - 10)³ is f⁻¹(x) = (∛x + 10) / 2.

Common Mistakes to Avoid

When finding inverse functions, there are a few common mistakes that students often make. Being aware of these pitfalls can help you avoid them and ensure you arrive at the correct answer.

  • Confusing f⁻¹(x) with 1/f(x): It's crucial to remember that f⁻¹(x) represents the inverse function, not the reciprocal of the function. These are two entirely different concepts.
  • Incorrectly swapping x and y: Swapping x and y is the cornerstone of finding the inverse. Make sure you swap them correctly before attempting to solve for y. A simple way to check is to rewrite the original equation with y as a function of x, then literally exchange the positions of the 'x' and 'y' variables.
  • Algebraic errors while solving for y: Solving for y often involves multiple algebraic steps. A small mistake in any of these steps can lead to an incorrect inverse function. Be meticulous and double-check your work at each step.
  • Forgetting to consider the domain and range: The domain of f(x) becomes the range of f⁻¹(x), and vice versa. It's important to consider these restrictions when defining the inverse function. In some cases, you might need to restrict the domain of the original function to ensure its inverse is also a function.

Verifying the Inverse Function

To ensure you've found the correct inverse function, you can verify your answer using the following property: f(f⁻¹(x)) = x and f⁻¹(f(x)) = x. This means that if you plug the inverse function into the original function, or vice versa, you should get x as the result.

Let's verify our inverse function, f⁻¹(x) = (∛x + 10) / 2, for f(x) = (2x - 10)³.

  1. f(f⁻¹(x)) = x f((∛x + 10) / 2) = (2((∛x + 10) / 2) - 10)³ = (∛x + 10 - 10)³ = (∛x)³ = x

  2. f⁻¹(f(x)) = x f⁻¹((2x - 10)³) = (∛((2x - 10)³) + 10) / 2 = (2x - 10 + 10) / 2 = 2x / 2 = x

Since both compositions result in x, we can confidently say that we have found the correct inverse function.

Conclusion

Finding the inverse of a function is a crucial skill in mathematics. By following the steps outlined in this article – replacing f(x) with y, swapping x and y, solving for y, and replacing y with f⁻¹(x) – you can confidently find the inverse of functions like f(x) = (2x - 10)³. Remember to avoid common mistakes and verify your answer to ensure accuracy. Understanding inverse functions opens the door to deeper mathematical concepts and problem-solving techniques. Keep practicing, and you'll master this essential skill!

For further exploration and practice, you might find helpful resources on websites like Khan Academy, which offers comprehensive math tutorials and exercises.