Find the Domain of y=-(x-3)²-1: Complete Function Analysis

To solve this problem, we'll follow these steps:

• Step 1: Identify the vertex of the parabola.
  The function $y = -\left(x-3\right)^2-1$ is in vertex form, with vertex $(h, k) = (3, -1)$.
• Step 2: Determine the shape and orientation of the parabola.
  Since $a = -1$ is negative, the parabola opens downwards.
• Step 3: Evaluate the function at the vertex.
  At $x = 3$, $y = -\left(3-3\right)^2 - 1 = -1$.
• Step 4: Analyze the function values as $x$ moves away from the vertex.
  Since the parabola opens downwards, $y$ decreases from the vertex value of $-1$, meaning $y < 0$ for all other $x \neq 3$.

Thus, the function is negative (less than zero) for all $x$. There are no $x$ values for which $y$ is positive (greater than zero).
In conclusion:

• Negative domain: all $x$
• Positive domain: none

Therefore, the solution matches choice 4:

Result:
$x < 0 :$ none
$x > 0 :$ all $x$