You have some text-form expression difficulties. You seem to want to find the inverse of
f(x) = [tex]\frac{1}{x+1}[/tex] - [tex]\frac{2}{9}[/tex]
In which you may create the transformation using f(x)=y of
x = [tex]\frac{1}{y+1}[/tex] - [tex]\frac{2}{9}[/tex]
Solve for y which will really be your inverse of the original function. Watch your algebra steps carefully, and then try checking your result again.
That looks very bad when I view the forum result; the 'tex' tagged portions look blackened and are not readable or legible. Is this just me here, or does it look that way for everybody?