I am stuck on this problem in my book for finding the domain and range of composite functions
For problems 8a-e I used a developed method to solve for the implied domain of these functions which produced correct results
Steps were:
- Use the domain of the first function and the range of the other trig function contained inside it
- Find the intersection of these two values
- substitute the contained trig function between the intersection value to find the domain
- substitute the domain values into the function to find the range
However, once I got to "8f" I found that the method I was using for the previous questions didn't carry over into arctan problems.
I see now that I probably have a fundamental misunderstanding of the process needed to solve these forms of problems. I'm guessing that question 10 is also related to this type of question
Could you please help explain the solving process for these kind of problems to me?
Thanks
It's not your fault that at least some of the questions are clearly incorrectly framed. In particular, 10 seems to be missing the inverse function on each of its outside $\sin$ or $\cos$ terms.
On 8f, you would want to note that $\cos x\in[-1,1]$ for all real numbers, and then identify the portion of the range of $\tan^{-1}$ corresponding to those values, i.e. find $\tan^{-1}([-1,1])$. Let me know if that gets you where you need to go!