Find the Domain of (x+3¼)² + ¾: Complete Function Analysis

To solve this problem, we need to determine when the function $y = \left(x + 3\frac{1}{4}\right)^2 + \frac{3}{4}$ is positive or negative across its domain.

The given function is in the vertex form $(x - h)^2 + k$, where $h = -3\frac{1}{4}$ and $k = \frac{3}{4}$. The parabola opens upwards because the coefficient of the squared term is positive.

The vertex of the parabola is at $(-3.25, 0.75)$. This means the minimum value of $y$ is $0.75$, which means $y$ is always greater than zero; the function does not reach zero or negative values.

Therefore, the function is positive for all $x$ and non-negative globally. There are no values of $x$ for which the function is negative.

Thus, for this function, the positive domain is all $x$.

Based on this analysis:

$x < 0 :$ none

$x > 0 :$ for all $x$