Frank Day

2022-07-09

Find the inverse function of
$f\left(x\right)=x-\left(2\sqrt{x}\right)+1$
the domain of definition is $\phantom{\rule{thinmathspace}{0ex}}x\ge 0$

Gornil2

Expert

So $y=x-2\sqrt{x}+1=\left(\sqrt{x}-1{\right)}^{2}$
$x=\left(1±\sqrt{y}{\right)}^{2}=1±2\sqrt{y}+y$
so ${f}^{-1}\left(x\right)=x±2\sqrt{x}+1$
Specifically ${f}^{-1}\left(x\right)=x+2\sqrt{x}+1$ when $x\ge 1$ and ${f}^{-1}\left(x\right)=x-2\sqrt{x}+1$ when $0\le x\le 1$.

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