Question

How do you verify that `f(x)=1/3x^3-2; g(x)=root3(3x+6)` are inverses?

Answer 1

`f(x)` and `g(x)` are inverses if `f(g(x))=x` and `g(f(x))=x`

Explanation:

If `color(blue)(f(x))=color(blue)(1/3x^3-2)`
and `color(red)(g(x))=color(red)(root(3)(3x+6))`

then
`color(white)("XXX")color(blue)(f(color(red)(g(x))))=color(blue)(1/3color(red)(g(x))^3-2)`

`color(white)("XXX")=color(blue)(1/3color(red)((root(3)(3x+6)))^3-2)`

`color(white)("XXX")=color(blue)(1/3 * color(green)((3x+6)) -2`

`color(white)("XXX")=color(green)(x+2) color(blue)(-2)`

`color(white)("XXX")=x`

~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~

and
`color(white)("XXX")color(red)(g(color(blue)(f(x)))=color(red)(root(3)(3color(blue)((1/3x^3-2))+6)`

`color(white)("XXX")=color(red)(root(3)(color(green)(x^3-6)+6)`

`color(white)("XXX")=color(red)(root(3)(color(green)(x^3))`

`color(white)("XXX")=x`