Stuck on functional eq/cannot find right substitution

Question

Find all functions $g: \mathbb{R} \to \mathbb{R} $ that satisfy:

$g(y)+g(\frac{y-1}{y})=\frac{5y^2-y-5}{y}$

In the past for problems like this, I usually try a bunch of substitutions(I have noticed prior problems usually have two varaiables, but this does not). I let y = 1 and got

$g(1) + g(0) = -1$

Then I tried y = 2

$g(2) + g(\frac{1}{2}) = \frac{13}{2}$

This led nowhere. Then I looked at some other posts, and defined this:

$f(y) = \frac{y-1}{y}$

and I found that:

$f(f(f(y))) = y$

So now I can say: $ g(f(y))=-g(y)+\frac{5y^2-y-5}{y}$

I think I am on the correct track, but not sure how to continue. Thank you for looking!

Answer 1

Hint: if $E_1$ is your equation, $E_2$ the equation with $y$ replaced by $f(y)$, and $E_3$ the equation with $y$ replaced by $f(f(y))$, consider $E_1 - E_2 + E_3$.