Proving a function is odd Here is the given question


  Show that 
  $$ f(x) = \frac{1}{2} - \frac{1}{2^x+1}$$
  is an odd function


I do know that 


  $ f(x) = f(-x)$ is a even function
  $ f(x) = -f(-x)$ is a odd function


But i cannot seem to simplify f(x) into -f(x) to prove that it is a odd function 
-f(x) is $ -\frac{1}{2} + \frac{1}{2^x+1}$ 

So far i have been able to get 


  $$\frac{\frac{1}{2}+2^x-2^{x+1}}{2+2^{x+1}}$$


and cannot find a way further to simplify
Note that 

$$\frac{1}{2^x+1}+\frac{2^x}{1+2^x}=1$$
Therefore
$$f(x)=\frac{1}{2}-\frac{1}{2^x+1}=-\frac{1}{2}+\frac{2^x}{2^x+1}=-\frac{1}{2}+\frac{1}{2^{-x}+1}=-f(-x)$$
hence $f(x)$ is odd.
$$2f(x)=\frac{2^x-1}{2^x+1}=\frac{1-2^{-x}}{1+2^{-x}}=-\frac{2^{-x}-1}{2^{-x}+1}.$$