Given a function f(x) with the domain `{x|-2<x<4,x in RR}` state the domain of `y=-2f(3(x+1))-5`

Note the transformations on f(x): the -2 performs a vertical stretch of factor 2 and a reflection over the horizontal axis. Neither of these transformations affects the domain. (They affect the range.)

The 3 performs a horizontal compression of factor 1/3. Thus the original domain is shrunk by a factor of 1/3 to `-2/3<x<4/3`

The 1 performs a horizontal translation (shift) left 1 unit. This shifts the domain 1 unit to the left. The new domain `-2/3-1<x<4/3-1`

** But `-2/3-1=-2/3-3/3=-5/3` and `4/3-1=4/3-3/3=1/3` so we can write the new domain as `-5/3<x<1/3` **

The -5 performs a vertical translation of 5 units down -- this only affects the range, not the domain.

Therefore, the new domain is `{x|-5/3<x<1/3,x in RR}`