Question

How do you find the inverse of `f(x)=x^2-2x-8` and is it a function?

Answer 1

The inverse of `f(x)=x^2-2x-8` is
`color(white)("XXX")bar(f)(x)=1+-sqrt(x+9)`
which is not a function since it generates multiple solution values.

Explanation:

If `color(red)(bar(f)(x))` is the inverse of `f(x)`
then `f(color(red)(bar(f)(x)))=color(blue)(x)` (by definition of inverse)

Therefore
`color(white)("XXX")(color(red)(bar(f)(x)))^2-2*(color(red)(bar(f)(x)))-8 = color(blue)(x)`

`color(white)("XXX")color(red)(bar(f)(x))^2-2color(red)(bar(f)(x)color(orange)(+1) =x+8color(orange)(+1)`

`color(white)("XXX")(color(red)(bar(f)(x))-1)^2=x+9`

`color(white)("XXX")color(red)(bar(f)(x))-1 = +-sqrt(x+9)`

`color(white)("XXX")color(red)(bar(f)(x))=1+-sqrt(x+9)`