Explanation:

To understand why option 'D' is the correct answer, let's first understand what a multiplexer (MUX) is and how it can be used to obtain different functions.

Multiplexer (MUX):
A multiplexer is a combinational logic circuit that selects one of several input signals and forwards it to a single output line. It has two sets of input lines: one set of data inputs (D0, D1, ..., Dn-1) and another set of control inputs (S0, S1, ..., Sn-1). The control inputs determine which data input is selected and forwarded to the output.

Function of a MUX:
The function of a MUX can be determined by the number of control inputs it has. For an n-bit MUX, it has 2^n input lines and n control inputs. The number of control inputs determines the number of inputs that can be selected.

8 : 1 MUX:
An 8 : 1 MUX has 8 input lines (D0, D1, ..., D7) and 3 control inputs (S0, S1, S2). It selects one of the 8 input lines based on the control inputs and forwards it to the output.

Functions of 3 Variables:
To obtain all functions of 3 variables (A, B, C), we need 2^3 = 8 different functions. Each function can be represented by a truth table with 3 inputs and 1 output.

All functions of 3 variables:
Using an 8 : 1 MUX, we can implement all 8 functions of 3 variables. We can connect each input line (D0, D1, ..., D7) to a specific output based on the truth table of the function we want to implement. The control inputs (S0, S1, S2) can be used to select the appropriate input line.

Functions of 4 Variables:
To obtain all functions of 4 variables (A, B, C, D), we need 2^4 = 16 different functions. Each function can be represented by a truth table with 4 inputs and 1 output.

Some but not all functions of 4 variables:
With an 8 : 1 MUX, we cannot directly implement all 16 functions of 4 variables. This is because there are not enough input lines (8) to represent all the possible combinations of the 4 variables. However, we can still implement some of the functions by using the available input lines and control inputs creatively.

For example, we can implement the functions where the output is solely based on the value of one input variable (A, B, C, or D). In this case, we can connect the input lines corresponding to the selected variable to the output. The remaining input lines can be connected to ground or any other fixed value.

Therefore, an 8 : 1 MUX can be used to obtain all functions of 3 variables and some but not all functions of 4 variables.