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**domain and range of the function help needed.. work shown!!!**

Given the function f(x)= (3x+4)/(5-2x) find the domain and range of the inverse function of f(x)

work shown:

it was proven to be one to one since f(x1)= f(x2)

found inverse f(x) to be (5x-4)/(3+2x), i believe this is correct

therefore the domain of the inverse function is the range of f(x)

therefore the range of the inverse function is the domain of f(x)

therefore the range of the inverse function is (-inf, 5/2), (5/2, inf)

since the domain of f(x) can't be 5/2 since 5-2x cannot equal 0 therefore x=0

i believe i may have made some incorrect assumptions for the range of the inverse function correctly

however what is the range of the inverse function, without graphically determining it

and is my domain for the inverse function correct

please help me with this problem