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!