Question

How do you find `[fog](x)` and `[gof](x)` given `f(x)=1+x` and `g(x)=x^2+5x+6`?

Answer 1

`[fog ] (x)=x^2+5x+7`
`[ gof ] (x) =x^2+7x+12`

Explanation:

This is a composition of functions

`f(x)=1+x`

`g(x)=x^2+5x+6=(x+2)(x+3)`

`[ fog ] (x)=f(g(x))=f(x^2+5x+6)`

`=1+x^2+5x+6=x^2+5x+7`

`[ gof ] (x)=g(f(x))=g(1+x)=(x+2+1)(x+3+1)`

`=(x+3)(x+4)`

`=x^2+7x+12`

`[ fog ] (x) != [ gof ] (x)`