Inverse CDF for Scaled Normal Distribution

I have a cumulative distribution function of the form:

This CDF can easily be written in MATLAB using the normcdf command as:

F = @(x) normcdf((a*x-b)/g(x),0,1);

I want to create a function handle for the inverse CDF (ICDF). If a=1 and g(x) is a constant, then the ICDF can simply be written as:

F_inv = @(Pr) norminv(Pr,b,g(x));

However neither of these conditions apply for me. One solution for creating a function handle for the ICDF is to invert F using the fzero() function:

F_inv = @(Pr) fzero(@(x) F(x)-Pr,x0); % x0 is an interval (a guess) where F(x)-Pr crosses the line y=0.

This however seems clunky and wont work if the guess for x0 does not contain the crossing point...is there a better way to do this?