Domain Analysis: Find Valid Inputs for y=-(x+√13)² Function

To find the positive and negative domains of the function, let's first analyze the given function $y = -\left(x + \sqrt{13}\right)^2$.

• Step 1: Analyze the function.
  The function is a downward-opening parabola with vertex at $(-\sqrt{13}, 0)$, because the coefficient of the quadratic term is negative.

• Step 2: Determine the intervals for positive and negative domains.
  A parabola that opens downward from its vertex means the function is negative for all $x \neq -\sqrt{13}$ since there are no values of x that make the function greater than 0 because the vertex is the maximum point.

• Step 3: Consider the function around the vertex.
  The only point where the function equals zero is at the vertex $x = -\sqrt{13}$. Thus, for any other $x$, the function is y < 0 .

Conclusion:
For x < 0 , the function satisfies the negative characteristic for all domains except $x = -\sqrt{13}$ where $y = 0$. Thus, the negative domain is x < 0 : x \neq -\sqrt{13} .

There are no values of $x$ for which the function becomes positive. Therefore, the positive domain is empty for x > 0 .

The solution concludes that the positive domain is none, and the negative domain is x < 0 : x \neq -\sqrt{13} .