Question

which best expresses the inverse of $f(x)=(x-1)^2+2$○ $f^{-1}(x)=\sqrt{x-2}-1$○ $f^{-1}(x)=(x+1)^2-2$○ $\sqrt{x+2}+1$○ $\sqrt{x+2}+1$

Explanation:

Step1: Replace $f(x)$ with $y$

$y=(x-1)^2+2$

Step2: Swap $x$ and $y$

$x=(y-1)^2+2$

Step3: Isolate the squared term

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

Step4: Take square root of both sides

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

Step5: Solve for $y$

$y=\sqrt{x-2}+1$
(Note: There appears to be a duplication/typo in the original options; the correct inverse is $\sqrt{x-2}+1$, which is not listed exactly as written, but the first option has a sign error, and the others use $x+2$ incorrectly. Assuming a typo in the options, the intended correct form is derived as above.)

Answer:

(Note: Correct inverse is $\boldsymbol{f^{-1}(x)=\sqrt{x-2}+1}$; none of the provided options match perfectly, but if we assume a typo in the options, the closest intended correct form is this expression. The given options contain errors: the first has a wrong sign, others use $x+2$ incorrectly.)