I'm not sure the relationship between transfer function from the 3dB point. The transfer function should give the gain, and I suppose when we put in the frequency at the 3dB point, the transfer function should be equal to a certain fraction?
No, you cannot calculate the transfer function given one point on its bode plot namely the 3dB point. How would you know what it did at DC or HF - you don't know if it's a low pass, high pass or bandpass filter. You can't even tell how many orders the filter has.
Here are a low pass and a high pass filter having the same 3dB point as an example: -
If all you measured was the 3dB point you wouldn't know which filter it was. Here is a bandpass filter: -
Which 3dB point are you measuring or have knowledge of?