From op amps 101-A, the formula for the gain of an inverting amp is - R (feedback) / R (input), however I'm wondering exactly how R (input) is determined in the first place.

R (input) is always between the input device (such as a transducer) and the summing point. However, if the transducer has its own resistance, it seems like that resistance would be R (input) and another resistor in series would not be required.

My guess is that R (input) might be for impedance matching between the transducer and the amp.

 

If there was no input resistor, the transducer would be operating into a short. The transducer might not like that?

 

The value for R (input) is typically selected so that its resistance is much higher than the source resistance. Of course this value is limited by input bias and frequency response considerations.
If the source has a significant output resistance than that could be used as R (input) for the purposes of determining the gain (assuming Mike's concern is not a factor).

Alternately, for a significant source resistance, you can use the op amp in a non-inverting follower configuration where the op amp (+) input has a very high impedance, meaning the gain would be little affected by the source resistance.

The value of R (source) normally has nothing to do with impedance matching. Impedance matching is typically done for high frequency transmission lines and where maximum power transfer is desired. Those conditions are not usual for op amp circuits.

 

As the previous two posters indicated, Rinput is the input impedance of the inverting opamp configuration with non-inverting input grounded. That is because the inverting input is at the same potential as the non-inverting input.

It is common practice to make Rinput at least 10-times the source resistance to minimize loading the source.

 

Again the correct universal answer is "it depends." One may pick a large impedance (resistance) so the source is not loaded down, one may pick one small to match a line impedance.
There are also frequency effects to consider as the input resistor forms an RC break with the capacitance of the pin on the amp. For very high frequency work the R may need to be very small; one 3MHz amp I designed had a 39 ohm (not K, ohm) input resistor. Do realize the current in the feedback network must come from somewhere and that somewhere is the input source and/or the amp's own output.
Generally, as a starting point avoid very large (>100K ohms or so) or small (<500 ohms) resistors.