4 To 1 MUX - Figuring out the inputs
Start by looking at the equation for a 2-in MUX:

$$
\begin{align*}
M(A,B,S) &amp;= A\cdot \bar{S} + B\cdot S
\end{align*}
$$

From this, you can derive some useful results (among others not listed here):

$$
\begin{align*}
1.~~~M(0,x,y) &amp;= x\cdot y \\
2.~~~M(x,0,y) &amp;= x\cdot\overline{y} \\
3.~~~M(x,y,0) &amp;= x \\
4.~~~M(1,x,y) &amp;= x + \overline{y} \\
5.~~~M(x,1,y) &amp;= x + y \\
6.~~~M(x,y,1) &amp;= y \\
7.~~~M(x,y,x) &amp;= x\cdot y \\
8.~~~M(x,y,y) &amp;= x + y \\
\end{align*}
$$

Now, your equation is:

$$f = \overline{x}z + yz + \overline{x}y = \overline{x}\left(y+z\right) + xyz $$

And you already have your required mux template laid out as:







simulate this circuit &ndash; Schematic created using CircuitLab

You could use a number of different approaches. But let's just use our imagination only and work from that.

The last mux, where \$x\$ is the selection input, selects between the following two options:

$$\begin{array}{rl}
\overline{x}&amp;z+y \\
x &amp; yz
\end{array}$$

Well, if you look up at the table of "useful results" above, you can easily see how to implement these.

For \$\overline{x}\$, you see from the table that either #5 or #8 would do the job here. Depending on what you feel is appropriate for your answer, pick one or the other.

For \$x\$, you can see that either #1 or #7 will do the job. Again, depending on what you feel is appropriate for your answer, pick one or the other.

For me? To reduce fanout, I'll pick the ones that have constants. So #1 and #5. Here's the result:







simulate this circuit

That's it, really. Sometimes, it helps you to just lay out a useful table for the mux function.