Transistor in the NOT Gate
How can the "input" be supplied with x volts, and the "output" be supplied with y volts? 

This is actually one of the useful functions of a transistor - it can be configure to use a small signal from a tiny switch or from the output of a micro-controller to switch a much larger load that the original switch could not.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. A transistor used as a simple power amplifier.

... how can you have 0 volts on the wire going out and 2.9 on the wire coming in?

It depends on the configuration. In Figure 1 when the switch is closed (pulling the input high) the transistor will turn on pulling its collector low. This is effectively an inverter.

If we are manipulating the voltage to manipulate whether the transistor functions as an open switch or a closed switch, why have a transistor?


Power gain. (See Figure 1 again.)
Voltage level changing. (e.g. 3 V to 24 V.)
Frequency of operation. The transistor can switch at far higher frequencies than a mechanical contact.
No moving parts on the transistor and no arcing. Let the mechanical switch handle signal currents which won't cause an arc and let the transistor handle the high currents without worry about arcing.