find the domain and range of
g(x) = {x + 3, x< -3
x^2+6, x> =-3

Domain is the numbers you can use
see we have rom x<-3 and x≥-3
therefor it is all real numbers for the domain

domain=all real numbers

and the range is the output
for all numbers less than -3, we have x+4, so -10+3=-7, so it extends infinetely in negative direction
and for x^2+6, you start with -3 which ends with (-3)^2+6=9+6=13, after that it gets bigger

so therefor the range is
from -∞ to -7, not including -7 and from 13 to positive infinity
in interval notation
(-∞,-7)U[13,∞)

domain=all real number
range=(-∞,-7)U[13,∞) or all real numbers except all numbers c, such that -7≤c<13