How do i show that this function :$f(x)=\alpha x +\beta+ \frac{\gamma}{\alpha'+\beta' x}$ always have symetric point?

Question

let $f$ be a real valued function defined as $f(x)=\alpha x +\beta+ \frac{\gamma}{\alpha'+\beta' x}$ , I want a simple method to show that every function of the precedent form always has a symetric center point ?

I have tried to show that using this basic : $f(2\alpha-x)=2\beta-f(x)$ but i didn't succeed , However this basic always work , And always the symetric point in this case is $M( -\frac{\alpha'}{\beta'}, f( -\frac{\alpha'}{\beta'}))$

Answer 1

It suffices to note $f\left(x-\frac{\alpha^{\prime}}{\beta^{\prime}}\right)=\alpha x+\frac{\gamma}{\beta^{\prime}x}+\beta-\frac{\alpha\alpha^{\prime}}{\beta^{\prime}}$ is an odd function plus a constant.