op amp single supply with nonlinear gain below 1 and start offset
You start with a non-inverting differential amplifier with 0V reference then increase the gain when Vout =Vth of transfer function and then increase the gain to linearize the input.  Since the non-inverting gain is always 1+ Rf/Rin,  you have two choices; { if Vout > x , increase Rf or if Vin >y decrease Rin.

For Rin, Depending on the voltage error tolerance of linearity you have options to use a precision switched resistance or a std  Zener or better, a precision Zener IC with a series R,  all in parallel with the fixed Rin to increase the gain.

When the threshold of the Zener is relatively low resistance , Zzt compared to the fixed Rin, the knee is as sharp as you need it to match the input response.