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

$f(x)=\frac{1}{x-2}$

Find out the inverse

Let

y=f(x)

$y=\frac{1}{x-2}$

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

$x=\frac{1}{y-2}$

isolate the variable y

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

Part 2

Verify the identity function

$(\text{fof}^{(-1)})=\frac{1}{(\frac{1}{x}+2)-2}$

simplify

$(\text{fof}^{(-1)})=\frac{1}{\frac{1}{x}}=x$

and

$(f^{(-1)}of)=\frac{1}{\frac{1}{x-2}}+2$

simplify

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

therefore

$(f^{(-1)}of)=(fof^{(-1)})$