Question

Let $f\left(x\right)=x^2-2x-1\forall x\in R$. Let $f:(-\infty ,a]\to [b,\infty )$, where $a$ is the largest real number for which $f\left(x\right)$ is bijective.
Let $f:[a,\infty )\to [b,\infty )$, then $f^{-1}\left(x\right)$ is given by

• A. $1+\sqrt{x+2}$
• B. $1-\sqrt{x+3}$
• C. $1-\sqrt{x+2}$
• D. $1+\sqrt{x+3}$

Answer 1

The correct option is A $1+\sqrt{x+2}$

$y=x^2-2x-1$

$y=(x-1{)}^2-2$

$\sqrt{y+2}=x-1$

$x=1+\sqrt{y+2}$

Hence $f^{-1}\left(x\right)=(1+\sqrt{x+2})$