Question

How would you solve these questions?

Answer 1

These functions exists (and are invertible) as long as the denominators are not zero: so we want

`x-a\\eq 0 \iff x\\eq a`

for
`f(x)`, and

`x-b\\eq 0 \iff x\\eq b`

for
`g(x)`

Now we show that they are inverses of each other: we want to show that

`f(g(x))=g(f(x))=x`. Let's start with
`f(g(x))=x`. We have

`f(g(x))=(1)/(g(x)-a)+b=(1)/((1)/(x-b)+a-a)+b=(1)/((1)/(x-b))+b=x-b+b=x`

And in the very same way we show that
`g(f(x))=x`