= Solution
At the final <steady state>, the <ligands> control activity as well as the amount of each factor. Assuming active X exceeds its thresholds and the high Y level exceeds $K_{yz}$, the normalized output production rates are
$$
\boxed{\begin{array}{cc|cc}
S_x&S_y&\text{I1 output rate}&\text{I4 output rate}\\
0&0&0&0\\
0&1&0&0\\
1&0&\beta_{\rm high}&\beta_{\rm low}\\
1&1&\beta_{\rm low}&\beta_{\rm low}
\end{array}}
$$
For <I1 incoherent feed-forward loop>, active X makes Y abundant, but Y can repress Z only if its own activating <ligand> is present. Without $S_y$, Y is inactive and the output remains high. For <I4 incoherent feed-forward loop>, active X represses Y to below its threshold, so the output is low regardless of $S_y$. If X is inactive, the direct permissive arm is absent and both outputs are zero even when Y is active.
Thus <I1 incoherent feed-forward loop> can retain two-input combinatorial control at <steady state> as well as generate the transient in part (a). <I4 incoherent feed-forward loop> loses steady-state sensitivity to $S_y$ once X has suppressed Y, under these particular assumptions. That gives a functional reason the first motif may be more useful in many regulatory contexts. It is not a proof of an evolutionary frequency: <promoter> implementation, thresholds, noise, costs and other functions also influence how common a motif is. The argument follows the supplied rate table, including its nonzero $\beta_{\rm low}$, rather than a generic Boolean AND truth table.
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