What is the minimum size of multiplexer needed to implement any boolean function of n variables if we are given a multiplexer and an inverter to use?
The less efficient (and maybe more obvious) way is to connect fixed logic levels to each input of the multiplexer. You need a 2^N input multiplexer for N inputs. So, an 16-input mux for any function of 4 bits. 

If, instead, you connect the respective inputs to either one of the inputs (that doesn't go to the multiplexer select input) or its complement, as required for the function, then you can halve the multiplexer size. So an 8-input mux (plus an inverter) for any function of 4 bits.