Please help me to find the range of .
Thanks in advance.
The range of is . Likewise, the range of is .
Since and are positive even powers, both have range .
BELOW is where others may disagree with me:
I contend that the range of is as opposed to , which the sum of the parts may intuitively suggest.
The two ranges differ insofar does not contain , whereas does contain .
I believe that the range of should NOT include 0, i.e., it should be .
Proof is achieved if we show on the entire domain (reals that are not multiples of ).
Both terms in are non-negative in the reals, so clearly the sum itself is non-negative.
cannot equal 0 because then either (which per the line above is not possible) OR , which also can't happen because and 0 cannot be a denominator.
As such, .
And with this new information, .
It at long last follows that the range of is .
If someone has an argument for a left bracket instead of my left open parenthesis, I'd love to hear it!
, evidently. My way was a fun adventure; I wanted to do it using identities for some bizarre reason. These problems nearly always boil down to the rudimentary sines and cosines. I see the airtight solution now, the symmetry, no plus 2 times "something positive". In retrospect, why the hell did I accept 2 as my best lower bound? Oh well, my Ravens are in the Super Bowl! Usually never post if I'm uncertain.