Show that the $f(x)=\frac{1}{2} +\frac{1}{2^x+1}$

Question

I am struggling with this question. Seems simple enough right but NO.
I might be just overthinking it, but someone please help. To know if it is an odd function if $$f(-x)=-f(x).$$

Answer 1

The Function $$f(x)=\frac{1}{2}+\frac{1}{2^x+1}$$ is not odd, you can check this by replacing $x$ by $-x$ and also by graph (it does not pass through origin, continuous at origin, $f(0)$ also defined). $$f(-x)=\frac{1}{2}+\frac{1}{2^{-x}+1}=\frac{1}{2}+\frac{2^{x}}{2^{x}+1} \not=-f(x)$$

But if you have done a typo, which you probably have- $$f(x)=-\frac{1}{2}+\frac{1}{2^x+1}$$ Then this is a odd function, take LCM. $$f(x)=\frac{1-2^x}{2(2^x+1)}$$ $$-f(x)=\frac{2^x-1}{2(2^x+1)}$$ $$f(-x)=\frac{1-2^{-x}}{2(2^{-x}+1)}=\frac{1-\frac{1}{2^x}}{2(\frac{1}{2^x}+1)}=\frac{2^x-1}{2(2^x+1)}=-f(x)$$ Also you can look at the graph.