Calculate Negative Area of f(x) = x² - 4: Below X-Axis Region

Find the negative area of the function

$f(x)=x^2-4$

To determine the interval where the function $f(x) = x^2 - 4$ is negative, follow these steps:

• Step 1: Identify the roots of the function by solving $x^2 - 4 = 0$.
  $x^2 = 4$ yields $x = 2$ and $x = -2$.
• Step 2: Consider the intervals created by these roots: $(-\infty, -2)$, $(-2, 2)$, and $(2, \infty)$.
• Step 3: Determine the sign of $f(x)$ in each interval by selecting a test point from each:
• For $x = 0$ in $(-2, 2)$:
  $f(0) = 0^2 - 4 = -4$ (Negative)
• For $x = -3$ in $(-\infty, -2)$:
  $f(-3) = (-3)^2 - 4 = 9 - 4 = 5$ (Positive)
• For $x = 3$ in $(2, \infty)$:
  $f(3) = 3^2 - 4 = 9 - 4 = 5$ (Positive)
• Step 4: The function $f(x)$ is negative only in the interval $(-2, 2)$.

Consequently, the interval where the function has negative values is $-2 < x < 2$, which aligns with choice 2 in the provided options.