Question

8 multiple choice 1 point given $f(x) = -sqrt{x - 2} + 5$, what is the domain and range of $f^{-1}(x)$? options: domain: all real numbers range: $y geq 5$; domain: $x geq 2$ range: $y leq 5$; domain: $x leq 2$ range: $y geq 5$; domain: $x leq 5$ range: $y geq 2$

Explanation:

Step1: Find domain of $f(x)$

For $\sqrt{x-2}$ to be real, $x-2\geq0$, so $x\geq2$.

Step2: Find range of $f(x)$

$\sqrt{x-2}\geq0$, so $-\sqrt{x-2}\leq0$, thus $f(x)=-\sqrt{x-2}+5\leq5$.

Step3: Relate to inverse function

The domain of $f^{-1}(x)$ is the range of $f(x)$, and the range of $f^{-1}(x)$ is the domain of $f(x)$.

Answer:

Domain: $x \leq 5$ Range: $y \geq 2$