Find the Domain of y=-3⅓x²-0.4: Complete Function Analysis

Find the positive and negative domains of the following function:

$y=-3\frac{1}{3}x^2-0.4$

To solve this problem, we examine the function $y = -\frac{10}{3}x^2 - 0.4$.

Step 1: Identify $a = -\frac{10}{3}$, which is negative. This means the parabola opens downwards.

Step 2: Find the zeros by setting $y = 0$:

$-\frac{10}{3}x^2 - 0.4 = 0$. Solving gives $-\frac{10}{3}x^2 = 0.4$, yielding

$x^2 = -\frac{0.4 \times 3}{10}$, which is negative. Therefore, no real solutions exist for zero crossing.

Step 3: Recognize since it doesn't cross the x-axis, the entire parabola is below the x-axis.

Conclusion: For $x < 0$, $y$ is negative for all $x$; for $x > 0$, $y$ remains negative.

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