How to find transfer function from a graph
Since the input is not a Heaviside Step function, we cannot presume this is a frequency-domain problem but rather a non-linear function of gain and offset.
Not knowing where this sketch comes from, let me assume it a quasi-linear so that the gain exists above some threshold and is saturated or null gain elsewhere. Consider tolerances as loose as the sketch for now.
Examine the Vout vs Vin and Avg to obtain gain and offset.
Using coarse estimates
t (ms) 0 to 25 to 50 +   Linear Mean
in     0 to  1  to 2      1.5
out    0 to  0  to 2      1  
gain   0 to  0==2==0 
offset = out-in         -0.5

gain=2   (for in = 1 to 2, otherwise =0)
offset = -0.5*2=-1  

**out = 2*(in-1)=in*2-2  (for in = 1 to 2)**

(rev B)

When amplifiers saturate, the linear gain =0 in this case for in&lt;1 and in&gt;2.  But in the linear out range, the gain ~ 2.
The soft edges mean there is a compressed range where the gain changes from 0 to 2 to 0.
In this plot, there is no observable filter except on the previous input signals.
A simulation.