Symmetry

[keetah_stockton]

I am good on the f(x)=f(-x), but I am having issues with symmetry about the origin. For example, one problem was x/x^2-6. I did not think there was any symmetry because all of the signs were not opposite. f(-x) would be -x/x^2-6, but the prof. sent it back as -(x/x^2-6) how is that possible, if you distribute the negative, it would make the six positive? Thanks in advance for your help.

 

[pka]

This always true: \(\displaystyle \L
- \left( {\frac{a}{b}} \right) = \left( {\frac{{ - a}}{b}} \right) = \left( {\frac{a}{{ - b}}} \right)\)

So if \(\displaystyle \L
f(x) = \frac{x}{{x^2 - 6}}\quad \Rightarrow \quad f( - x) = \frac{{ - x}}{{\left( { - x} \right)^2 - 6}} = \frac{{ - x}}{{x^2 - 6}} = - \frac{x}{{x^2 - 6}} = \frac{x}{{ - x^2 + 6}}\).