What is the effect of manipulation of a function over its domain range and graph?

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Suppose I have a function f of x another function G of x which is derived by manipulating f of x. Domain of f is D1 and its range is R1 whereas domain of G is D2 and range is R2. Now what can we said about domains, ranges and graphs. Will they be exactly same or almost same with except at some points. Please make this clear. EDIT1: https://pasteboard.co/ISG3HPa.jpg Attaching this image over my analysis, please tell is the conclusion is correct.