Question

How do you show that `f(x)=1-x^3` and `g(x)=root3(1-x)` are inverse functions algebraically and graphically?

Answer 1

See the explanation below

Explanation:

If a function `f(x)` has an inverse `f^-1(x)` then

The composition

`f(f^-1(x))=x`

and `f^-1(f(x))=x`

Here, we have

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

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

Then

`f(g(x))=f(root(3)(1-x))=1-(root(3)(1-x))^3=1-(1-x)=x`

`g(f(x))=g(1-x^3)=root(3)((1-(1-x^3)))=root(3)(x^3)=x`

Therefore,
algebraically `f(x)` and `g(x)`are inverse functions

graph{(y-1+x^3)(y-root(3)(1-x))(y-x)=0 [-3.117, 3.813, -0.884, 2.58]}

Graphically, the functions `f(x)` and `g(x)` are symmetric with

respect to the line `y=x`. So they are inverse functions.