https://file.aiquickdraw.com/mathbot/file/172667533701960e49f09.jpg

solve this problem and draw me the graph. and show me the steps also

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.

Sketch the graph and determine the domain and range of the following function: f(x) = {x + 9 if x < -2, |x - 2| + 1 if -2 ≤ x < 3, √(x - 2) if x > 3}.

Learn how to sketch the graph of the piecewise function f(x) and determine its domain and range. This step-by-step guide explains key concepts such as linear functions, absolute value functions, and square root functions. Ideal for students in Grades 9-12.

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