finding composition of functions, the set A is the set of real numbers EXCLUDING zero.

the function f is injective and defined by f(x) = 1 - 1/x  for all x in the set A

need to show that f of f of f is equal to i subscript A, not sure what is meant by i subscript A
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1 Answer

f(x)=1-(1/x)=(x-1)/x.

f(f(x))=(f(x)-1)/f(x)=

((x-1)/x-1)/((x-1)/x)=

((x-1-x)/x)/((x-1)/x)=

-1/(x-1).

f(-1/(x-1))=((-1/(x-1))-1)/(-1/(x-1))=

((-1-x+1)/(x-1))(-(x-1))=x.

Therefore, f(f(f(x)))=x.

[For example, if x=2, f(2)=½, f(½)=-1, f(-1)=2. So f(f(f(2)))=2=x.]

i_A may indicate a definition of i_A. But I think it means any element of set A. So the compound function maps each element onto itself. Zero is excluded because it would generate 1/0 which is undefined.

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