1. Stating the problem: We have the function $$y = \frac{4}{-x - 2} - 4$$ and need to find: 1.1 The domain of the function. 1.2 The roots (zeros) of the function. 1.3 The intervals where the function is positive. 2. Domain determination: The function is a rational function with denominator $$-x - 2$$. The denominator cannot be zero because division by zero is undefined. Set denominator equal to zero: $$-x - 2 = 0$$ Solve for $$x$$: $$-x = 2 \implies x = -2$$ So, the domain is all real numbers except $$x = -2$$. Domain: $$\{x \in \mathbb{R} \mid x \neq -2\}$$. 3. Finding roots (where $$y=0$$): Set the function equal to zero: $$0 = \frac{4}{-x - 2} - 4$$ Add 4 to both sides: $$4 = \frac{4}{-x - 2}$$ Multiply both sides by $$-x - 2$$: $$4(-x - 2) = 4$$ Simplify left side: $$-4x - 8 = 4$$ Add 8 to both sides: $$-4x = 12$$ Divide both sides by -4: $$x = -3$$ Check that $$x = -3$$ is in the domain (it is). So, the root is $$x = -3$$. 4. Finding where the function is positive ($$y > 0$$): Start with: $$\frac{4}{-x - 2} - 4 > 0$$ Add 4 to both sides: $$\frac{4}{-x - 2} > 4$$ Multiply both sides by $$-x - 2$$, but remember to reverse inequality if $$-x - 2 < 0$$. Consider two cases: Case 1: $$-x - 2 > 0 \implies -x > 2 \implies x < -2$$ Multiply inequality by positive number (no sign change): $$4 > 4(-x - 2)$$ Simplify right side: $$4 > -4x - 8$$ Add $$4x$$ and $$8$$ to both sides: $$4 + 4x + 8 > 0 \implies 4x + 12 > 0$$ Subtract 12: $$4x > -12$$ Divide by 4: $$x > -3$$ Combine with domain condition $$x < -2$$: $$-3 < x < -2$$ Case 2: $$-x - 2 < 0 \implies x > -2$$ Multiply inequality by negative number (reverse inequality): $$4 < 4(-x - 2)$$ Simplify right side: $$4 < -4x - 8$$ Add $$4x$$ and $$8$$ to both sides: $$4 + 4x + 8 < 0 \implies 4x + 12 < 0$$ Subtract 12: $$4x < -12$$ Divide by 4: $$x < -3$$ Combine with domain condition $$x > -2$$: No $$x$$ satisfies $$x > -2$$ and $$x < -3$$ simultaneously. Therefore, the function is positive only on the interval: $$(-3, -2)$$. Final answers: - Domain: $$\{x \in \mathbb{R} \mid x \neq -2\}$$ - Root: $$x = -3$$ - Positive values: $$x \in (-3, -2)$$