Hooke's Law Explanation & Practical Applications

I wanted to have on discussion on this problem..
When we are compressing airin a cylinder using a cylinder-piston arrangement, the density of air changes and hence the force transmitted by it will also change..
This means that air in uncompressed state will transmit less force whereas air in compressed state wikk transmit more.. That is the basics of pneumatic systems. Is this correct or not??
Also can someone help me in finding the formula for the change in density and change in force transmitted that takes place.

Within the elastic limit of a solid material, the deformation, produced by a force of any kind is proportional to the force.

Because this observation and the corresponding Hooke's Law formula will hold true only upto corresponding stress-strain value as defined by the Elastic limit on the Stress-Strain curve.

This law can be applied only upto the point where the body regains its orignal dimensions after load is removed or atleast the body can be assumed to have regined its original dimension for all practical purposes - holds true for elastic limit

If you load the body beyond elastic limit the body will undergo significant plastic deformation, meaning the body will not regain its original dimensions even after the the load is removed - you dont want your body to get deformed

Hence, in order to ensure that the Law remains valid, we need to specify the limiting condition, "Within the elastic limit"

Actually it is more 11th standard Physics than Engineering 😉 But it is a critical Engineering design no doubt