 #1
 236
 21
 Homework Statement:

Find the domain and range of
##f(x) = 10^{\sin x} + 10^{\csc x}##
 Relevant Equations:
 ##D_f = {D_h}\cap {D_g}##
## Let~~f(x)=h(x)+g(x) , where~~h(x)=10^{\sin x}~~and~~g(x)=10^{\csc x}##
##Then,~~D_f = {D_h}\cap {D_g}##
##Clearly,~~D_h=ℝ~~and~~D_g=ℝ\{nπn∈ℤ\}##
##∴~~D_f =ℝ\{nπn∈ℤ\}##
After considering the new domain, the range of ##\sin x## in ##10^{\sin x}## is ##[1,1]\{0\}##
Therefore, the range of ##10^{\sin x}## shall be ##[0.1,10]\{1\}##
The range of ##\csc x## in ##10^{\csc x}## is ##{(∞,1]}\cup{[1,∞]}##
Therefore, the range of ##10^{\csc x}## is ##{(0,0.1]}\cup{[10,∞)}##
As you can see, the ranges of h(x) and g(x) are totally 'opposite'. How to find the range of f(x)?
P.S. I tried Wolfram Alpha and it didn't work.
##Then,~~D_f = {D_h}\cap {D_g}##
##Clearly,~~D_h=ℝ~~and~~D_g=ℝ\{nπn∈ℤ\}##
##∴~~D_f =ℝ\{nπn∈ℤ\}##
After considering the new domain, the range of ##\sin x## in ##10^{\sin x}## is ##[1,1]\{0\}##
Therefore, the range of ##10^{\sin x}## shall be ##[0.1,10]\{1\}##
The range of ##\csc x## in ##10^{\csc x}## is ##{(∞,1]}\cup{[1,∞]}##
Therefore, the range of ##10^{\csc x}## is ##{(0,0.1]}\cup{[10,∞)}##
As you can see, the ranges of h(x) and g(x) are totally 'opposite'. How to find the range of f(x)?
P.S. I tried Wolfram Alpha and it didn't work.