How do I calculate the duty cycle range for DC-DC buck converter?
Here is a hint: -


  Both switching devices will be MOSFETs


This tells me that the buck converter is a synchronous type with a push-pull type switch arrangement - this makes estimating the duty cycle very easy; one switch connects the output to the incoming positive rail then the other switch grounds the output. The average voltage of the switched waveform for an input voltage of 3.7 volts and an output voltage of 1.2 volts is ......? Please fill-in the gap

If you don't quite understand where I'm coming from consider the "proper" output after the inductor and capacitor and think of what the inductor is - it passes DC i.e. if the input voltage has an average value of X then the output voltage also has an average value of X (yes there will be a small volt drop due to current thru the inductor because of the inductor's dc resistance but this can be largely ignored for a decent approximation).