Question

For the function `f(x)=(x-3)^3+1`, how do you find `f^-1(x)`?

Answer 1

In order to find for any function `f(x)`, we must apply the 'transformation' `y=x`. In order to do that, we must define `x` in terms of `y`, ie., find `f(y)`, then set `y=x`.

Let `y=f(x)`

`y=(x-3)^3+1`

`y-1=(x-3)^3`

`x-3=root(3)(y-1)`

`x=3+root(3)(y-1)`

We've now found `f(y)`, so we must set `y=x` by replacing `x` with `y` and `y` with `x`.

`f^-1(x)=y=3+root(3)(x-1)`