what is the domain and range of1) [x] [ ] represents greatest integer function2) {x} { } represents fractional part

The domain of both greatest integer function and fractional part is xelement of R because they can take all the values of x.

The [x], range is the entire set of integers (that is the definition of a greatest integer function)
Say for n≤x<n+1, then [x] = n

So for [f(x)], you need to find the range of f(x) for the given domain of f(x). If domain is not given, then you need to identify the domain of f(x) first, and then in that domain you need to find the range of f(x).
Once range of f(x) is found, then range of [f(x)], is obtained by splitting the range into intervals n≤f(x)<n+1.
For fractional part of x, the range is 0≤{x}<1.

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