Domain Analysis: Finding Valid Inputs for y=-(x+8/9)²

To solve the problem, we follow these steps:

• Step 1: Recognize that the function $y = -\left(x + \frac{8}{9}\right)^2$ is a quadratic function in vertex form where the leading coefficient $a = -1$.
• Step 2: Determine the direction of the parabola. Since the coefficient $a$ is negative, the parabola opens downwards.
• Step 3: Analyze the function's vertex: The vertex is at $x = -\frac{8}{9}$. At this point, $y = 0$, which is the maximum value.
• Step 4: Consider the behavior of the function on either side of the vertex. Because of the downward opening, $y$ is negative for all $x$ except at the vertex itself.
• Step 5: Conclude the results: The function is negative for all $x$, except $y = 0$ at $x = -\frac{8}{9}$.

Thus, the positive and negative domains are:

$x < 0 : x \ne -\frac{8}{9}$ (negative domain)

$x > 0 :$ none (positive domain)

The correct answer choice is:

$x < 0 : x \ne -\frac{8}{9}$

$x > 0 :$ none