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.
