Question 927655: please solve the symmetry and odd even or neither for the following equation:
x^3 - 9x / 4x^2 - 4x - 80
Answer by jim_thompson5910(35256)   (Show Source):
You can put this solution on YOUR website! Let 
To prove an function is odd, we need to show that  . So let's first find 
Now let's find 
So we can see that  (notice how the +4x and -4x in the denominators don't match).
Therefore,  is NOT an odd function.
------------------------------------------------------------------------------------------
To show that an function is even, we need to show that 
From the last part, we found that but this is not equal to .
The numerators don't match at all in terms of their signs and the denominators are off too a bit (again the +4x and -4x).
Therefore, is NOT an even function.
------------------------------------------------------------------------------------------
Summary: Above we've shown that is neither even nor odd.
Because of this, it does NOT have y axis symmetry (it's not even) and it does NOT have origin symmetry (it's not odd).
The graph visually sums it all up for us without having to say a single word. It looks like the graph has origin symmetry, but it's approximately symmetric for  . Any x outside this interval has this symmetry breaking down.
From  to  , this graph looks pretty symmetric (with respect to the origin), but as x goes beyond +4, the curve does not go beyond -4 to reciprocate which is where the symmetry starts to break down.
A more basic and quick indication is to notice how the vertical asymptotes  and  are not equal in magnitude. Ie,  , so this also tells us f(x) does not have origin symmetry.
------------------------------------------------------------------------------------------------------------------------
If you need more one-on-one help, email me at jim_thompson5910@hotmail.com. You can ask me a few more questions for free, but afterwards, I would charge you ($2 a problem to have steps shown or $1 a problem for answer only).
Alternatively, please consider visiting my website: http://www.freewebs.com/jimthompson5910/home.html and making a donation. Any amount is greatly appreciated as it helps me a lot. This donation is to support free tutoring. Thank you.
Jim
------------------------------------------------------------------------------------------------------------------------
|
|
|
