Determine X Values for y = -x² - 4 Less than Zero

Given the function:

$y=-x^2-4$

Determine for which values of x the following is true:

$f\left(x\right) < 0$

To solve this problem, we'll follow these steps:

• Step 1: Analyze the quadratic function given: $y = -x^2 - 4$.
• Step 2: Analyze what $y < 0$ implies for the values of $x$.

Now, let's solve it:

Step 1: Consider the given quadratic function $y = -x^2 - 4$. The function $y = -x^2$ represents a downward-opening parabola due to the negative sign before $x^2$. The entire graph of the parabola, being shifted downward by 4 units with the term $-4$, will be wholly beneath the x-axis since there's no positive vertex or value. The vertex of the parabola is at $(0, -4)$, which is already below zero.

Step 2: Because the quadratic term causes the graph to be a parabola opening downwards, it means that for any $x$, $y = -x^2 - 4 \leq -4$, which is always less than zero. Thus, the inequality $f(x) = -x^2 - 4 < 0$ is satisfied for all real numbers $x$.

Therefore, the solution is that the inequality is true for all $x$.