Question

the function $y = \sqrt{x}$ is translated using the rule $(x, y) \to (x - 7, y + 2)$ to create $f(x)$. what is the domain of $f(x)$?
$\\{x \mid x \geq -7\\}$
$\\{x \mid x \geq 7\\}$
$\\{x \mid x \geq -2\\}$
$\\{x \mid x \geq 2\\}$

Explanation:

Step1: Find translated function

The translation rule $(x,y)\to(x-7,y+2)$ means we replace $x$ with $x-7$ in the original function and add 2 to the output. So $f(x)=\sqrt{x-7}+2$.

Step2: Set radicand non-negative

For square roots, the expression inside must be $\geq0$:
$x-7 \geq 0$

Step3: Solve for x

Add 7 to both sides:
$x \geq 7$

Answer:

$\{x | x \geq 7\}$