The question is: f(x)=√(x+1) Show that (f·f^(-1))(a)=a=(f^(-1)·f)(a) for any a in the respective domains. Can somebody explain the question to me? I'm not even sure what it…

Question

The question is:

f(x)=
`√(x+1)`
Show that (f·
`f^(-1)`)(a)=a=(
`f^(-1)`·f)(a) for any a in the respective domains.

Can somebody explain the question to me? I'm not even sure what it means. What are respective domains??

Answer 1

`f(x) = √(x+1) \\ y = √(x+1) \ or \ y^2 = x+1 \ or \ x = y^2 - 1 \\ Therefore, \ f^(-1)(x)=x^2 - 1`


`(fof^(-1))(a) = f(f^(-1)(a))=f(a^2-1) \\ =√(a^2-1+1)=√(a^2)=a \\ \\ (f^(-1)of)(a)=f^(-1)(f(a))=f^(-1)(√(a+1))= \\ (√(a+1))^2-1 = a+1-1 = a`

Therefore,
`(fof^(-1))(a) = (f^(-1)of)(a) = a`