For the solutions written in the second image, the range of fg is done correctly.

Range of g =[1,∞ )

Domain of f =(0, ∞ )

Range of g intersecting Domain of f = [1,∞ )

This interval [1,∞ ) becomes the new domain for the graph of f, ie

the range corresponding to the domain of [1,∞ ) for the function f = [2,∞ )

Therefore, [2,∞ ) is the range of fg.

**This approach definitely works for any question. **

Hope this helps. Peace.