Definition: An inverse function reverses another function: if f sends x to y, then f inverse sends y back to x, and the two graphs mirror each other across y = x.
What is an inverse function?
If f takes an input and produces an output, the inverse, written f-1, takes that output and returns the original input. So f-1(f(x)) = x. Only one-to-one functions, which pass the horizontal line test, have an inverse that is also a function.
How do you find it?
- Write y = f(x).
- Swap x and y.
- Solve for y.
Worked example: f(x) = (5x – 3)/2. Write y = (5x – 3)/2, swap to x = (5y – 3)/2, then 2x = 5y – 3, so 5y = 2x + 3 and f-1(x) = (2x + 3)/5.
Check: f(1) = 1, and f-1(1) = 5/5 = 1.
What is the common confusion?
The -1 is not an exponent. f-1(x) does not mean 1/f(x). Mathbot keeps the two ideas apart: a reciprocal flips a fraction, an inverse swaps the roles of input and output.
Example: f(x) = (5x – 3)/2 has inverse (2x + 3)/5
Where to learn it
- Combining and Inverting Functions – lesson
- Domains, Composition and Inverse Functions Explained – lesson