We have the function #f(x)=2^(-x)#
The domain is the set of all possible #x#-values which will make the function "work", and will output real #f(x)#-values.
So in the given function any real number value of #x# will output real #f(x)# values.
Therefore the domain is #RR# (all real values).
In interval notation it can be written as #x in (-oo,+oo)#
The range is the resulting #f(x)#-values we get after substituting all the possible #x#-values
We know #x# can be any real number.
#color(red)1.#So let #x=# a negative number.
Then the function becomes #f(x)=2^(-(-x))#
#f(x)=2^x# which will always be a positive number.
#color(red)2.#Now let #x=# a positive number.
Then the function becomes #f(x)=2^(-x)# which can be written as #f(x)=1/2^x#.
Now as #x# increases #f(x) -> 0# but #f(x)# will never be #0#.
From this we can conclude that #f(x)# can only be a positive number inbetween #0# and #+oo# but cannot be #0#.
Therefore #f(x) in (0,oo)# and we use this ( ) bracket because neither #0# or #oo# are included in the values of #f(x)# but all the values in between #0# and #oo# are included in the values of #f(x)# aka the Range.
