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

Video Solution

Solution Steps

00:00 Find the positive domain of the function

00:03 Looking at the coefficient of X squared, it's negative so the function is concave down

00:13 The positive domain is actually above the X-axis

00:17 Therefore we'll set Y=0 to find the intersection points with X-axis

00:22 Isolate X

00:31 Take the square root

00:34 When taking a square root there are 2 solutions (positive and negative)

00:39 These are the intersection points with the X-axis

00:51 Let's mark the intersection points with the X-axis

01:05 The negative domain is below the X-axis

01:09 The positive domain is above the X-axis

01:13 Let's find the domain where the function is positive

01:18 And this is the solution to the question

Step-by-Step Solution

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

$-x^2 + 9 = 0$

$x^2 = 9$

$x = \pm 3$

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)

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 $-3 < x < 3$ .