- #1

baldbrain

- 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-\{nπ|n∈ℤ\}##

##∴~~D_f =ℝ-\{nπ|n∈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)?

##Then,~~D_f = {D_h}\cap {D_g}##

##Clearly,~~D_h=ℝ~~and~~D_g=ℝ-\{nπ|n∈ℤ\}##

##∴~~D_f =ℝ-\{nπ|n∈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.*