https://file.aiquickdraw.com/mathbot/file/17273032847161qqwnng4.jpg

The function provided in the image is:

\[
g(x) = 3 - \sqrt{6 - 2x}
\]

This is a square root function. To understand its behavior and domain, let's break it down:

1. **Domain of the function**:
   - The square root function \(\sqrt{6 - 2x}\) requires that the expression inside the square root must be non-negative (since the square root of a negative number is not real in standard real-valued functions).
   - Therefore, \(6 - 2x \geq 0\).
   
   Solving for \(x\):
   \[
   6 - 2x \geq 0 \quad \Rightarrow \quad -2x \geq -6 \quad \Rightarrow \quad x \leq 3
   \]
   So, the domain of the function is \(x \in (-\infty, 3]\).

2. **Range of the function**:
   - The square root function \(\sqrt{6 - 2x}\) always gives non-negative values.
   - The largest value of \(6 - 2x\) occurs when \(x = -\infty\), which tends towards positive infinity.
   - Therefore, \(g(x) = 3 - \text{some non-negative value}\), and the function will decrease from 3 downwards, depending on \(x\).

Would you like to proceed with any of the following analyses?

1. Analyze the function's derivative to find critical points.
2. Plot the graph of \(g(x)\) for visualization.
3. Find the behavior of the function as \(x\) approaches 3 from the left.
4. Discuss possible transformations based on the square root.
5. Explore further transformations or possible simplifications.

**Tip:** Whenever you deal with square roots in functions, always check the domain first to avoid undefined values!

Understand how to find the domain and range of the square root function g(x) = 3 - √(6 - 2x). Learn about square root restrictions and how to solve inequalities for function domains. Suitable for high school students in Grades 9-12.

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