Question

What is the inverse of f(x) = -1/5x -1 ?

Answer 1

`f(y) = (y-1)/(5y)`

Explanation:

Replace `f(x)` by `y`

`y = -1/(5x-1)`

Invert both sides

`1/y= -(5x-1)`

Isolate `x`

`1-1/y = 5x`
`1/5-1/(5y) = x`

Take the least common divisor to sum the fractions

`(y-1)/(5y) = x`

Replace `x` for `f(y)`

`f(y) = (y-1)/(5y)`

Or, in `f^(-1)(x)` notation, replace `f(y)` for `f^(-1)(x)` and `y` for `x`

`f^(-1)(x) = (x-1)/(5x)`

I personally prefer the former way though.

Answer 2

`g(x) = -5x-5`
is the inverse of `f(x)=-1/5x-1`

Explanation:

If `g(x)` is the inverse of `f(x)`
then `f(g(x)) = x`

Replacing `x` with `g(x)` in the original equation and
recognizing that `f(g(x)) = x`
we have
`color(white)("XXX")f(g(x)) = -1/5g(x)-1=x`

`rarrcolor(white)("XXXXXXXX")-1/5g(x) =x+1`

`rarrcolor(white)("XXXXXXXX")g(x) = (-5)(x+1) = -5x-5`