Here is the graph of the piecewise function $f(x)$ along with the steps to find the domain and range:

Domain:
The domain is $(-\infty, \infty)$, as the function is defined for all real numbers.

Range:
To determine the range, we analyze each part of the function:

1. For $x < -2$, $f(x) = x + 9$ is a linear function that decreases without bound as $x \to -\infty$.
2. For $-2 \leq x < 3$, $f(x) = |x - 2| + 1$. The minimum value occurs at $x = 2$, where $f(2) = 1$, and it increases symmetrically away from this point.
3. For $x > 3$, $f(x) = \sqrt{x - 2}$. The smallest value of this part is $f(3) = 1$, and as $x \to \infty$, $f(x)$ increases without bound.

Thus, the range of the function is $(-\infty, \infty)$, as the function covers all real values.

Would you like more details or have any questions?

Here are 5 related questions to further explore:

1. How would the graph change if we modified the function to $f(x) = |x - 1| + 1$ for the middle piece?
2. What happens if we restrict the domain of the piecewise function to $x > 0$?
3. How does the graph behave if we use $f(x) = -(x + 9)$ for $x < -2$?
4. Can you calculate the derivative of the piecewise function at $x = -2$?
5. What would happen if the square root function is replaced by $(x - 2)^2$ for $x > 3$?

Tip: For piecewise functions, it's important to carefully examine how each piece connects at the boundaries to ensure continuity or identify discontinuities.