Domain Analysis of y=-(x-11)²: Finding Valid Input Values

Let's analyze the problem by rewriting the function in its vertex form:

The given function is $y = -\left(x - 11\right)^2$.

Step 1: Identify the vertex and parabola direction.

• The vertex is $(11, 0)$ meaning at $x = 11$, the value of $y$ is zero.
• Since the parabola opens downwards (as the coefficient of $(x - 11)^2$ is negative), the output of the function will always be negative except at the vertex where it is zero.

Step 2: Determine the positive and negative domains of the function.

• The entire real number line minus the vertex point is where the function value is negative.
• The function never achieves positive values, so the positive domain is essentially non-existent. Therefore, all $x \neq 11$ fall into the negative domain.

Thus, the positive and negative domains of the function are:

$x < 0 : x\ne11$

$x > 0 :$ none

Hence, the solution is, the function is negative for all values except at $x = 11$, where it is precisely zero.

The correct choice according to our analysis is Choice 2:

$x < 0 : x\ne11$

$x > 0 :$ none