Please help me to find the range of .

Thanks in advance.

Printable View

- Feb 3rd 2013, 06:37 AMkjchauhanRange
Please help me to find the range of .

Thanks in advance. - Feb 3rd 2013, 07:19 AMPlatoRe: Range
- Feb 3rd 2013, 11:05 AMabenderRe: Range

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 insofarcontain , whereas**does not**contain .**does**

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!

-Andy - Feb 3rd 2013, 11:46 AMPlatoRe: Range
- Feb 3rd 2013, 12:21 PMabenderRe: Range
, 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.