Inverse of Nonconstant Linear Functions

Fix $b \in \mathbb{R}$. For all $z \in \mathbb{R}$ such that $z \neq -1, 0, 1,$ define a function $f_z$ such that $f_z(x) = zx + b$.

Now define a function $g:\mathbb{R} \backslash \{0,1\} \to \mathbb{R}$ such that $g(z)$ is the unique value $c$ where $f_z(c) = f_z^{-1}(c)$.

Which of the following is a correct expression for $g(z)$?