Calculate Area Under f(x) = -x² + 9: Quadratic Function Analysis

Find the positive area of the function
$f(x)=-x^2+9$

To determine the positive area of the function $f(x) = -x^2 + 9$, we follow these steps:

• Step 1: Find the roots of the equation $-x^2 + 9 = 0$.
• Step 2: Solve the equation for the roots:

$-x^2 + 9 = 0$

$x^2 = 9$

$x = \pm 3$

• Step 3: Identify the intervals defined by these roots: $(-\infty, -3)$, $(-3, 3)$, and $(3, \infty)$.
• Step 4: Test the sign of $f(x)$ in each interval:

For interval $(-3, 3)$: Choose $x = 0$, $f(0) = 9$ (positive)

For interval $(-\infty, -3)$: Choose $x = -4$, $f(-4) = -7$ (negative)

For interval $(3, \infty)$: Choose $x = 4$, $f(4) = -7$ (negative)

• Step 5: Conclude the positive area is for $-3 < x < 3$.

The positive area of the function is described by $-3 < x < 3$, which corresponds to the second choice.

Therefore, the solution to the problem is $-3 < x < 3$.