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

I need to verify the identities of two functions and find the inverse of a one-to-example-1

1 Answer

6 votes

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)))

User Brian D
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