Let f be a function with domain D and let g be an extension

Let f be a function with domain D, and let g be an extension of f with domain A. Then by definition, f = g_D and D A Let i be the inclusion mapping from D to A given by i(x) = x for all x D. Prove that f = g i.

Solution

To show that f=goi where f=g|D and D is in A, where D is domain of f and A is domain of g.

i is an identity funciton defined by i(x)=x for all x in D.

the domain and range of the function i is D, then goi range is also equal to range of f.

Let x in D,then f(x)=g(x)=g(i(x))=goi(x)

Therefore, f=goi