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

1 Answer
Jul 19, 2016

Answer:

#-oo < x < +oo#
#-oo < f(x) < 2#

Explanation:

Function #y = 2^x# is defined for all real #x#.
It is always positive.
It is monotonically increasing .
If #x# goes to #-oo#, it asymptotically decreasing to #0#.
If #x# goes to #+oo#, it increasing to #+oo#.

Therefore, function #y=-2^x# has the following properties:
It is defined for all real #x#.
It is always negative.
It is monotonically decreasing .
If #x# goes to #-oo#, it asymptotically increasing to #0#.
If #x# goes to #+oo#, it decreasing to #-oo#.

Next step is to add #2# to this function to get #y=2-2^x#.
It has the following properties:
It is defined for all real #x#.
It is always less than #2#
It is monotonically decreasing .
If #x# goes to #-oo#, it asymptotically increasing to #2#.
If #x# goes to #+oo#, it decreasing to #-oo#.

We might suggest to use educational materials on UNIZOR.COM to study functions and their behavior. Exponential functions are explained there in lots of detail.