Solve y = x²-4: Finding Values Where Function is Negative

To determine for which values of $x$ the function $f(x) = x^2 - 4$ is less than 0, we need to solve the inequality $x^2 - 4 < 0$.

First, find when the function equals zero by solving $x^2 - 4 = 0$. This gives the roots as $x^2 = 4$, or $x = \pm 2$, specifically $x = 2$ and $x = -2$.

Next, perform a sign analysis of $f(x) = x^2 - 4$ in the intervals defined by these roots: $(-\infty, -2)$, $(-2, 2)$, and $(2, \infty)$.

• For $x \in (-\infty, -2)$: Pick a test point, such as $x = -3$. Calculate $f(-3) = (-3)^2 - 4 = 9 - 4 = 5$. Here, the function is positive.
• For $x \in (-2, 2)$: Pick a test point, such as $x = 0$. Calculate $f(0) = 0^2 - 4 = -4$. Here, the function is negative, satisfying $f(x) < 0$.
• For $x \in (2, \infty)$: Pick a test point, such as $x = 3$. Calculate $f(3) = 3^2 - 4 = 9 - 4 = 5$. Here, the function is positive.

Thus, the function $f(x) = x^2 - 4$ is less than 0 for $-2 < x < 2$.

The correct interval representing the solution is $-2 < x < 2$.