express the function in the form fog  F(x)=2x+x^2)^4 

Expert Answers
sciencesolve eNotes educator| Certified Educator

You should remember the definition of composition of functions, such that:

`(fog)(x) = f(g(x))`

Hence, `g(x)` replaces x in equation of `f(x)` , thus, you need to consider `f(x) = x^4` and `g(x) = 2x+x^2` .

Composing the functions `f(x)` and `g(x)` and replacing `g(x)` for x in equation of function yields:

`(fog)(x) = f(g(x)) => f(g(x)) = (g(x))^4`

Replacing the equation of `g(x)` in `f(g(x))` yields:

`f(2x+x^2) = (2x+x^2)^4`

Hence, you may write the function `F(x) = (2x+x^2)^4` as a composition of two functions `f(x) = x^4` and `g(x) = 2x+x^2` , such that `(fog)(x) = (2x+x^2)^4` .