Help with following question about Muxes?
Let's first list out the cases for the box marked with an (\$*\$). Per the text with your problem:

$$\begin{align*}
f_0&amp;= a \oplus b\\\\
f_1&amp;= \overline{a \oplus b}\\\\
f_2&amp;= a \cdot b\\\\
f_3&amp;= \overline{a \cdot b}
\end{align*}$$

Let's call the mux using \$c\$ and \$d\$ as select inputs \$m_0\$. The other mux with \$f\$ as its output will be called \$m_f\$. Then the two k-maps for the two muxes are:

$$\begin{array}{rl}
\begin{smallmatrix}\begin{array}{r|cc}
m_0&amp;\overline{c}&amp;c\\
\hline
\overline{d}&amp;b&amp;f_3\\
d&amp;a&amp;f_2\end{array}\end{smallmatrix}
&amp;
\begin{smallmatrix}\begin{array}{r|cc}
m_f&amp;\overline{m_0}&amp;m_0\\
\hline
\overline{f_0}&amp;f_0&amp;d\\
f_0&amp;f_1&amp;c\end{array}\end{smallmatrix}
\end{array}$$

Substituting the expressions for \$f_0\$ through \$f_3\$ into the above k-maps:

$$\begin{array}{rl}
\begin{smallmatrix}\begin{array}{r|cc}
m_0&amp;\overline{c}&amp;c\\
\hline
\overline{d}&amp;b&amp;\overline{a\, \cdot\, b}\\
d&amp;a&amp;a\, \cdot\, b\end{array}\end{smallmatrix}
&amp;
\begin{smallmatrix}\begin{array}{r|cc}
m_f&amp;\overline{m_0}&amp;m_0\\
\hline
\overline{a\,\oplus\, b}&amp;a\,\oplus\, b&amp;d\\
a\,\oplus\, b&amp;\overline{a\,\oplus\, b}&amp;c\end{array}\end{smallmatrix}
\end{array}$$

If you want, you could proceed through the necessary algebra, starting with:

$$m_0=b\,\overline{c}\,\overline{d}+a\,\overline{c}\,d+\overline{a\, b}\,c\,\overline{d}+a\, b\,c\,d$$

But that's more painful than it needs to be.

You only have 16 permutations. So just list them out in a table:

$$\begin{smallmatrix}
\begin{array}{cccc|cc}
a&amp;b&amp;c&amp;d&amp;m_0&amp;m_f\\
\hline
0&amp;0&amp;0&amp;0&amp; 0&amp;0\\
0&amp;0&amp;0&amp;1&amp; 0&amp;0\\
0&amp;0&amp;1&amp;0&amp; 1&amp;0\\
0&amp;0&amp;1&amp;1&amp; 0&amp;0\\
0&amp;1&amp;0&amp;0&amp; 1&amp;0\\
0&amp;1&amp;0&amp;1&amp; 0&amp;0\\
0&amp;1&amp;1&amp;0&amp; 1&amp;1\\
0&amp;1&amp;1&amp;1&amp; 0&amp;0\\
1&amp;0&amp;0&amp;0&amp; 0&amp;0\\
1&amp;0&amp;0&amp;1&amp; 1&amp;0\\
1&amp;0&amp;1&amp;0&amp; 1&amp;1\\
1&amp;0&amp;1&amp;1&amp; 0&amp;0\\
1&amp;1&amp;0&amp;0&amp; 1&amp;0\\
1&amp;1&amp;0&amp;1&amp; 1&amp;1\\
1&amp;1&amp;1&amp;0&amp; 0&amp;0\\
1&amp;1&amp;1&amp;1&amp; 1&amp;1
\end{array}
\end{smallmatrix}$$

That's a lot easier. From the above table it is trivial to develop the k-map for \$m_f\$ (remember that \$f=m_f\$):

$$\begin{smallmatrix}\begin{array}{r|cccc}
m_f&amp;\overline{a}\,\overline{b}&amp;\overline{a}\,b&amp;a\,b&amp;a\,\overline{b}\\
\hline
\overline{c}\,\overline{d}&amp;0&amp;0&amp;0&amp;0\\
\overline{c}\, d&amp;0&amp;0&amp;1&amp;0\\
c\, d&amp;0&amp;0&amp;1&amp;0\\
c\, \overline{d}&amp;0&amp;1&amp;0&amp;1
\end{array}\end{smallmatrix}$$

Obviously, the above results in:

$$\begin{align*}
m_f&amp;=a\,b\,d + \overline{a}\,b\,c\,\overline{d}+a\,\overline{b}\,c\,\overline{d}\\
&amp;=a\,b\,d +\left(\overline{a}\,b+a\,\overline{b}\right)c\,\overline{d}\\
&amp;=a\,b\,d +\left(a\oplus b\right)c\,\overline{d}
\end{align*}$$

You should be able to easily construct the necessary gates -- especially since the question allows you to assume ready access to both the inverted and non-inverted values of inputs \$a\$ through \$d\$.