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What is purpose of R1. How does it work. How does it bias it in linear region.

 

The logic inverter is a Cmos type with an extremely high input impedance. The resistor adds AC and DC negative feedback so the inverter becomes a linear amplifier.

 

It is somewhat similar to how an op amp works. If the input voltage goes up the output goes down, and visa versa. A balance is therefore reached somewhere in between.