Question

What is the inverse function?

Answer 1

If `f` is a function, then the inverse function, written `f^(-1)`, is a function such that `f^(-1)(f(x)) = x` for all `x`.

Explanation:

For example, consider the function:

`f(x) = 2/(3-x)`

(which is defined for all `x != 3`)

If we let `y = f(x) = 2/(3-x)`, then we can express `x` in terms of `y` as:

`x = 3-2/y`

This gives us a definition of `f^-1` as follows:

`f^(-1)(y) = 3-2/y`

(which is defined for all `y != 0`)

Then `f^(-1)(f(x)) = 3-2/f(x) = 3-2/(2/(3-x)) = 3-(3-x) = x`