Calculate the Positive Area: Finding Region Where x² - 4 > 0

To solve find the positive area of the function $f(x) = x^2 - 4$, we proceed as follows:

• Step 1: Find the x-intercepts by setting $f(x) = 0$:

$x^2 - 4 = 0$

$x^2 = 4$

Take the square root of both sides:

$x = \pm 2$

• Step 2: Identify the intervals formed by the intercepts:

The intercepts $x = -2$ and $x = 2$ divide the x-axis into three intervals: $(-\infty, -2)$, $(-2, 2)$, and $(2, \infty)$.

• Step 3: Test each interval to determine where $f(x) > 0$:

- For $x \in (-\infty, -2)$, pick $x = -3$:
$f(-3) = (-3)^2 - 4 = 9 - 4 = 5 > 0$, so the function is positive.

- For $x \in (-2, 2)$, pick $x = 0$:
$f(0) = (0)^2 - 4 = -4 \leq 0$, so the function is not positive.

- For $x \in (2, \infty)$, pick $x = 3$:
$f(3) = (3)^2 - 4 = 9 - 4 = 5 > 0$, so the function is positive.

Conclusively, the function $f(x)$ is positive in the intervals $x < -2$ and $x > 2$.

The correct answer is: $x < -2$ or $2 < x$.