how to find the domain and range of a COMPOSITE function?

Question

anonymous · Answer

I am really confused on how to get the final answer.  It would be really nice if i could get a step by step on how to do these problem.

anonymous · Answer

i know that for g(f(x)) the range/outputs of f(x) is going to be the input for g(x). But from here, im confused.

amistre64 · Answer

the domain of g(x) includes:  {-4,-3,-2,-1,0,1,2,3,4}

the range of f(x) that falls within the domain of g(x) affects what part of x we can use.

in this case all of the range of f(x) is contained in the domain of g(x)
 $$ \{-4,-3,\underbrace{(-2,-1,0,1,2)}_{range~f(x)},3,4\}$$

amistre64 · Answer

so the range of g(f) is defined for the outputs that are generated between f = -2 and 2|dw:1413053328573:dw|