Wiring outlet with a switch for garbage disposal
You don't have a 3-wire cable; you have two independent 2-wire cables. To have two independent receptacles of the duplex receptacle with two 2-wire cables both tabs need to be broken off, so the neutral side must be broken as well. The hot and the neutral wires connected to a given receptacle must be in the same cable.

If the switch does not control one of the hot conductors at the receptacle, then the switch may be faulty or more likely wired wrong. Look in the box for the switch and see how the switch is wired.

EDIT
The currently capped white neutral must be connected to the neutral side of the receptacle that it's companion hot is connected to. Break off the neutral tab if it is not already broken.