The natural logarithm is only defined if the value inside it is greater than #0#. This means that the domain of the function will be where the bit inside the logarithm is positive:
#xe^x+1>0#
#xe^x> -1#
We notice that #e^x# is always positive, so the only way for the left hand side to be negative is if #x# is negative. Note however that the function actually never goes more negative than #-1#. We can see this if we plug in some negative number, #-t#, into the function:
#-te^-t=-t/e^t#
Since #e^t>t#, the fraction will always be less than #1#, and therefor the expression can never produce a value smaller than #-1#.
This means that the inequality #xe^x> -1# holds for all #x in RR#, and that means that the function is defined for all real numbers, #x in RR#.
