Domain Analysis: Finding Valid Inputs for y = (1/6)x² - 5

We begin by finding the roots of the function $y = \frac{1}{6}x^2 - 5$ by setting it to zero:

$\frac{1}{6}x^2 - 5 = 0$

Multiply through by 6 to clear the fraction:

$x^2 - 30 = 0$

Solve for $x^2 = 30$, giving:

$x = \pm \sqrt{30}$

Now, we analyze the intervals determined by these roots:

• For $x \in (-\infty, -\sqrt{30})$, since the parabola opens upwards, $y$ is positive.
• For $x \in (-\sqrt{30}, \sqrt{30})$, the parabola dips below the x-axis, so $y$ is negative.
• For $x \in (\sqrt{30}, \infty)$, the parabola returns above the x-axis, making $y$ positive.

Therefore, the function is negative in the interval $-\sqrt{30} < x < \sqrt{30}$, and positive in the interval $x < -\sqrt{30}$ or $x > \sqrt{30}$.

This gives us the positive and negative domains:

x < 0 : -\sqrt{30} < x < \sqrt{30}

x > \sqrt{30} or x > 0 : x < -\sqrt{30}