Question

How would you find `f^-1` and verify that `f(f^-1(x))=f^-1(f(x))=x`, if `f(x)=2x+3`? `f(x)=x^3+1`?

Answer 1

`f(f^(-1)(x))= f^(-1)(f(x))`, if and only if f(x)=x.

Answer 2

I'll do `f(x) = x^3 + 1` and leave `f(x) = 2x + 3` up to you to do for practice.

The inverse of a function can be found algebraically by switching the values of `x` and `y` inside the function:

`y = x^3 + 1`

`x = y^3 + 1`

`x - 1 = y^3`

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

Let's now use your trick that this new function is indeed the inverse function.

`(root(3)(x - 1))^3 + 1 =^? x`

`x - 1 + 1 =^? x`

`x = x`

So, if `f(x) = x^3 + 1`, then `f^-1(x) = root(3)(x - 1)`

Hopefully this helps!