How-Tos / Math

How to Find the Inverse of a Function

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What is the Inverse of a Function?

The inverse of a function f(x) is another function, denoted as f-1(x), that undoes the action of f(x). In simpler terms, if you apply f(x) to a value x, and then apply f-1(x) to the result, you should get back to the original value of x.

Steps to Find the Inverse of a Function

  1. Replace f(x) with y: Start by replacing f(x) in the function expression with y. So, if you have a function f(x) = …, rewrite it as y = ….
  2. Swap x and y: Swap the variables x and y in the equation obtained from step 1. This step transforms the equation into the form of the inverse function.
  3. Solve for y: Rearrange the equation from step 2 to solve for y. This may involve isolating y on one side of the equation.
  4. Replace y with f-1(x): Once you’ve solved for y, replace it with f-1(x) to express the inverse function.
  5. Verify the Inverse: To ensure correctness, verify that f(f-1(x)) = x and f-1(f(x)) = x for all suitable values of x.

Example:

Let’s find the inverse of the function f(x) = 2x + 3

  1. Replace f(x) with y: y = 2x + 3.
  2. Swap x and y: x = 2y + 3.
  3. Solve for y: x – 3 = 2y, y = \frac{x-3}{2}.
  4. Replace y with f-1(x): f-1(x) = \frac{x-3}{2}.

Conclusion

Finding the inverse of a function involves a systematic approach, and with practice, it becomes more intuitive. Remember to check your solution by verifying the properties of inverses. Now that you’ve learned the method, go ahead and tackle those functions with confidence!