Question

I need to verify the identities of two functions and find the inverse of a one-to-one function

Answer 1

we have that

`f(x)=(1)/(x-2)`

Find out the inverse

Let

y=f(x)

`y=(1)/(x-2)`

Exchange the variables (x for y and y for x)

`x=(1)/(y-2)`

isolate the variable y

`\begin{gathered} y-2=(1)/(x) \\ y=(1)/(x)+2 \\ f^(-1)(x)=(1)/(x)+2 \\ \end{gathered}`

Part 2

Verify the identity function

`(\text{fof}^((-1)))=(1)/(((1)/(x)+2)-2)`

simplify

`(\text{fof}^((-1)))=(1)/((1)/(x))=x`

and

`(f^((-1))of)=(1)/((1)/(x-2))+2`

simplify

`\begin{gathered} (f^((-1))of)=x-2+2 \\ (f^((-1))of)=x \end{gathered}`

therefore

`(f^((-1))of)=(fof^((-1)))`