Domain Exploration of y = x² + 0.5x + 5: Positive and Negative Insights

To solve this problem, we follow these steps:

• Step 1: Determine if the function intersects the x-axis by finding the roots.
• Step 2: Compute the discriminant of the quadratic equation.
• Step 3: Analyze the parabola's direction and minimum point.

Now, let's work through each step:
Step 1: The given quadratic function is $y = x^2 + \frac{1}{2}x + 5$. The standard quadratic form is $ax^2 + bx + c$ where $a = 1$, $b = \frac{1}{2}$, and $c = 5$.
Step 2: To determine the roots, let's calculate the discriminant, $\Delta = b^2 - 4ac$.
For our function, $\Delta = \left(\frac{1}{2}\right)^2 - 4 \cdot 1 \cdot 5 = \frac{1}{4} - 20 = \frac{1}{4} - 20 = -\frac{79}{4}$.
Since the discriminant is negative, the quadratic has no real roots, indicating that it does not intersect the x-axis. Thus, it does not pass below the x-axis.

Step 3: Since $a = 1$ is positive, the parabola opens upwards. Since there are no real roots, it suggests that the function is always positive.

Therefore, the solution to the problem is that the function is positive for all $x$. There is no $x$ for which the function is negative, since it never crosses the x-axis.

Thus, the solution is:
$x > 0 :$ all $x$
$x < 0 :$ none

This corresponds to choice 4.