What is the domain and range of #(2/3)^x – 9#?

1 Answer
May 28, 2017

Answer:

Domain: #(-oo, oo)#
Range: #(-9, oo)#

Explanation:

First note that #(2/3)^x-9# is well defined for any Real value of #x#. So the domain is the whole of #RR#, i.e. #(-oo, oo)#

Since #0 < 2/3 < 1#, the function #(2/3)^x# is an exponentially decreasing function which takes large positive values when #x# is large and negative, and is asymptotic to #0# for large positive values of #x#.

In limit notation, we can write:

#lim_(x->-oo) (2/3)^x = -oo#

#lim_(x->oo) (2/3)^x = 0#

#(2/3)^x# is continuous and strictly monotonically decreasing, so its range is #(0, oo)#.

Subtract #9# to find that the range of #(2/3)^x# is #(-9, oo)#.

Let:

#y = (2/3)^x-9#

Then:

#y+9 = (2/3)^x#

If #y > -9# then we can take logs of both sides to find:

#log(y+9) = log((2/3)^x) = x log(2/3)#

and hence:

#x = log(y+9)/log(2/3)#

So for any #y in (-9, oo)# we can find a corresponding #x# such that:

#(2/3)^x-9 = y#

That confirms that the range is the whole of #(-9, oo)#.