center of symmetry formula

Question

How to prove that $I(0,-1)$ is the center of symmetry of the function
$$F(x)= x - \dfrac{2e^x}{(e^x -1)}$$ Is there any formula that I can directly apply?

Answer 1

Define $y(x) = F(x) + 1$. As y is shifted $1$ upwards its center should be $(0,0)$. Central symmetry in origin implies: $y(-x) = -y(x)$ You must demonstrate:

$F(-x) + 1 = -F(x) - 1$

Answer 2

For any point $I(a,b)$, prove that
$$f(a-x)+ f(x) =2b$$ In your case: $a=0$ and $b=-1$.
Then you have to prove $f(-x)+f(x) =-2$