Domain Analysis of y = (1/2)x² + 2x + 3: Finding Valid Inputs

To solve this problem, we'll analyze the given quadratic function $y = \frac{1}{2}x^2 + 2x + 3$ and determine where it is positive and where it is negative.

Step 1: Calculate the discriminant to find out if there are real roots.

The quadratic equation has coefficients $a = \frac{1}{2}$, $b = 2$, $c = 3$. The discriminant $\Delta$ is given by:

$\Delta = b^2 - 4ac = 2^2 - 4(\frac{1}{2})(3) = 4 - 6 = -2$

Since the discriminant is negative ($\Delta < 0$), the quadratic equation does not have real roots; thus, it does not cross the x-axis.

Step 2: Determine the nature of the parabola.

The parabola opens upwards because the leading coefficient $a = \frac{1}{2}$ is positive.

Step 3: Conclude based on the parabola's direction and lack of real roots.

Because the parabola opens upwards and does not intersect the x-axis, the function $y$ is positive for all $x$.

Therefore, the positive domain of the function is for all $x$, and there is no negative domain.

Conclusion:
$x > 0 :$ for all x
$x < 0 :$ none