SOLUTION: please solve the symmetry and odd even or neither for the following equation: x^3 - 9x / 4x^2 - 4x

SOLUTION: please solve the symmetry and odd even or neither for the following equation: x^3 - 9x / 4x^2 - 4x - 80

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.

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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.

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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.

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