If you remove the base biasing resistor from a transistor, how does it change its functionality?

If there's no biasing resistor, the transistor might not work at all during at least some of the input voltage sinusoidal swing, i.e., the transistor might not even be in its operational regime (and the Transistor Man equation would not apply). This might result in a seriously distorted or even non-existent output. You want to choose the base-biasing resistor(s) so that the base voltage is at least $V_{pn}$ above the emitter voltage, and you typically also want the collector voltage to be a few volts above the emitter voltage (and less than $V_{CC}$). There may be other requirements on output currents or voltages that you select the base resistor in order to meet.