Question

If f:A→B is a one to one onto function, then f⁻¹ o f= ____​

Answer 1

Step-by-step explanation:

f:A→Bf:A→B and g:B→Cg:B→C are both one-to-one functions.

Suppose a1,a2∈Aa1,a2∈A such that (gof)(a1)=(gof)(a2)(gof)(a1)=(gof)(a2)

⇒g(f(a1))=g(f(a2))⇒g(f(a1))=g(f(a2)) (definition of composition) Since gg is one-to-one, therefore,

f(a1)=f(a2)f(a1)=f(a2)

And since ff is one-to-one, therefore,

a1=a2a1=a2

Thus, we have shown that if (gof)(a1)=(gof)(a2)(gof)(a1)=(gof)(a2) then a1=a2a1=a2

Hence, gofgof is one-to-one function.

Answer 2

Answer:

f:A→Bf:A→B and g:B→Cg:B→C are both one-to-one functions.

Suppose a1,a2∈Aa1,a2∈A such that (gof)(a1)=(gof)(a2)(gof)(a1)=(gof)(a2)

⇒g(f(a1))=g(f(a2))⇒g(f(a1))=g(f(a2)) (definition of composition) Since gg is one-to-one, therefore,

f(a1)=f(a2)f(a1)=f(a2)

And since ff is one-to-one, therefore,

a1=a2a1=a2

Thus, we have shown that if (gof)(a1)=(gof)(a2)(gof)(a1)=(gof)(a2) then a1=a2a1=a2

Hence, gofgof is one-to-one function.