Find the Domain of (x+2.7)² + 0.4: Analyzing Function Inputs

To solve this problem, we need to determine the domains for the given function $y = (x + 2.7)^2 + 0.4$ where $y$ is positive and negative.

The function is a quadratic function in the form $y = (x + h)^2 + k$, representing a parabola opening upwards. The vertex of this parabola is at $x = -2.7$ and $y = 0.4$, meaning this point is the minimum point of the parabola.

The $y$-value of the function at its minimum is $y = 0.4$. Because the parabola opens upwards, it implies that for all $x$, $y \geq 0.4$.

Since the minimum value of $y$ is 0.4, the function never takes negative values; therefore, there is no negative domain.

The positive domain, $x > 0$, can be interpreted as being satisfied by all $x$, since no values make $y$ less than 0. The function's range is therefore always positive, including its minimum value.

Conclusively, the positive domain is all $x$, while the function has no negative domain.

Thus, the final solution is:

$x > 0 :$ all $x$

$x < 0 :$ none