The problem in the image is asking you to graph the function $h(x) = \sqrt{x - 2}$ by applying transformations to the base function $f(x) = \sqrt{x}$. It also asks to find the domain and range of the function.

Step-by-Step Solution:

1. Base Function: The basic function is $f(x) = \sqrt{x}$. The graph of this function starts at $(0,0)$ and increases to the right, being defined only for $x \geq 0$.

2. Transformation: The function $h(x) = \sqrt{x - 2}$ represents a horizontal shift of the base function $\sqrt{x}$ to the right by 2 units. The transformation here is $x - 2$, which shifts the graph 2 units to the right.

   • Shift: Every point on the graph of $\sqrt{x}$ moves right by 2 units.

3. Domain: Since the square root function $\sqrt{x - 2}$ is only defined when $x - 2 \geq 0$, we solve: $x - 2 \geq 0 \quad \Rightarrow \quad x \geq 2$ So, the domain is $[2, \infty)$.

4. Range: The square root function always produces non-negative values. So the range remains $[0, \infty)$.

Correct Graph:

The graph of $h(x) = \sqrt{x - 2}$ starts at $(2, 0)$ and rises to the right. This matches Option D from the multiple-choice options shown in the image.

Would you like more details or have any other questions?

Related Questions:

1. What happens to the graph of $\sqrt{x}$ if you replace $x$ with $x + 3$?
2. How does a vertical stretch affect the graph of $f(x) = \sqrt{x}$?
3. How do you find the domain of a general square root function $\sqrt{g(x)}$?
4. What is the effect of a vertical shift on a square root function?
5. How do you reflect the graph of a square root function across the x-axis?

Tip: For transformations, always remember that changes inside the function (with $x$) affect the graph horizontally, while changes outside affect it vertically.