Find the Domain of y=(x+2)²: Analyzing Positive and Negative Inputs

To solve the problem of finding the positive and negative domains of the function $y = (x + 2)^2$, we will explore the nature of this function.

Step 1: Given Function and Analysis
The function $y = (x + 2)^2$ is a quadratic function presented in vertex form. The vertex form of a quadratic function is generally given as $y = a(x - h)^2 + k$, but simplifying our function, this translates to $y = 1(x + 2)^2 + 0$, indicating a parabola that opens upwards with vertex at $x = -2$.

Step 2: Zero Point of the Function
The only point where $(x + 2)^2 = 0$ is when $x + 2 = 0$. Solving for $x$, we get $x = -2$. At this point, $y = 0$.

Step 3: Positive and Negative Analysis
Since the parabola opens upwards (as the coefficient 1 in front of $(x+2)^2$ is positive), the function will yield positive $y$ values for all $x$ except at the vertex $x = -2$. Thus, for $x \neq -2$, the function is always positive, and it is never negative.

Step 4: Conclusion
Therefore, since $(x + 2)^2$ can only yield zero at $x = -2$ and positive for every other real $x$, the domains are:
- For $y < 0$ (Negative domain): None
- For $y > 0$ (Positive domain): All $x \neq -2$, including positive numbers only for $x > 0$.

Therefore, the solution to the problem is:
$x < 0 :$ none
$x > 0 : x \ne -2$