How do you find the domain and range of #y=2^(-x)#?

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Answer 1 · 2017-05-09T06:31:54

The domain is #(-oo, oo)# and the range is #(0, oo)#.

Given:

#y = 2^(-x)#

First note that this is well defined for any real value of #x#.

So the domain is the whole of #RR#, or in interval notation #(-oo, oo)#.

Next note that #2^(-x) > 0# for any real value of #x#. So the range is #(0, oo)# or a subset of it.

Let #y > 0#. Then #log(y)# is well defined. So we can take logs of both sides to find:

#log(y) = log(2^(-x)) = (-x) log(2)#

Dividing both sides by #-log(2)# we find:

#x = -log(y)/log(2)#

So that tells us that for any #y > 0# there is an #x# such that:

#y = 2^(-x)#

So the range is the whole of #(0, oo)#.