Question

How do you find the inverse of `f(x) = x/(x+1)` and is it a function?

Answer 1

If `bar(f)(x)` is the inverse of `f(x)=x/(x+1)`
then
`color(white)("XXX")bar(f)(x)=x/(1-x)` (which is a function.)

Explanation:

If `color(red)(bar(f)(x))` is the inverse of `color(blue)(f(x))`
then
`color(white)("XXX")color(blue)(f"(")color(red)(bar(f)(x))color(blue)(")") = x`
and
`color(white)("XXX")color(blue)(f"(")color(red)(bar(f)(x))color(blue)(")")=(color(red)(bar(f)(x)))/(color(red)(bar(f)(x))+1)`

`color(white)("XXX")color(red)(bar(f)(x))=x*(color(red)(bar(f)(x))+1)`

`color(white)("XXX")color(red)(bar(f)(x))-x*color(red)(bar(f)(x)) =x`

`color(white)("XXX")color(red)(bar(f)(x))*(1-x)=x`

`color(white)("XXX")color(red)(bar(f)(x)) = x/(1-x)`

Since this expression provides a unique solution for all valid values of `x`; it is a function.