Question

What is the inverse of `f(x)=sqrt(x)+6`?

Answer 1

If `f(x)=sqrt(x)+6`
then `g(x)=x^2-12x+36` is the inverse of `f(x)`

Explanation:

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

...but we also have;
`f(g(x))=sqrt(g(x))+6` (by given definition of `f(x)`)

Therefore
`color(white)("XXX")sqrt(g(x))+6=x`

`color(white)("XXX")rarr sqrt(g(x))=x-6`

`color(white)("XXX")rarr g(x)=(x-6)^2=x^2-12x+36`

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Some people use the notation `f^(-1)(x)` for the inverse of `f(x)`.

I find this confusing since it conflicts with the more general usage of the notation `f^k(x)` meaning `[f(x)]^k`