The domain of a function is all possible values of #x #where #f(x)# is defined. Here, when the denominator is equal to 0, the function is undefined. In this case:
#x+6=0#
#x=-6#
So #y# is only undefined at #x=-6#. In interval notation, we write the domain as #(-oo,-6)uu(-6,oo)#.
The range of a function is all possible values for #y#. Another way to solve for the range is to find #y^-1#, as in, the inverse function of #y#, and find its domain. Here, we can find #y^-1#:
#y=3/(x+6)#
Switch the variables and solve for #y#:
#x=3/(y+6)#
#1/x=(y+6)/3#
#3/x=y+6#
#y=3/x-6#
This is #y^-1#. It is also defined when its denominator is equal to #0#.
So #y^-1# will be undefined when #x=0#, and therefore #y# will be undefined when #y=0#. The range in interval notation is:
#(-oo,0)uu(0,oo)#
