To find the inverse of a function, you essentially reverse the roles of the input (x) and the output (f(x) or y). Here's a step-by-step guide based on the reference:
Steps to Find the Inverse of a Function
Here's a structured way to find the inverse of a function, f(x):
- Rewrite f(x) as y: This makes the notation simpler to work with. So, replace f(x) with y.
- Switch x and y: This is the core step where you reverse the roles of input and output. Every instance of x becomes y, and every instance of y becomes x.
- Solve for y: Isolate y on one side of the equation. This will express y in terms of x.
- Rewrite y as f-1(x): Replace y with the proper notation for the inverse function, which is f-1(x). This signifies that you've found the inverse function.
Here is a table representing the steps:
| Step | Description | Example: Finding the inverse of f(x) = 2x + 3 |
|---|---|---|
| 1 | Rewrite f(x) as y | y = 2x + 3 |
| 2 | Switch x and y | x = 2y + 3 |
| 3 | Solve for y | x - 3 = 2y => y = (x - 3) / 2 |
| 4 | Rewrite y as f-1(x) | f-1(x) = (x - 3) / 2 |
Therefore, the inverse of f(x) = 2x + 3 is f-1(x) = (x - 3) / 2.
Example: Let's find the inverse of f(x) = x3.
- Step 1: y = x3
- Step 2: x = y3
- Step 3: y = ∛x
- Step 4: f-1(x) = ∛x
Therefore, the inverse of f(x) = x3 is f-1(x) = ∛x.
