Question

Consider the function f(x) =(9- x2)/(x2-4)

For which intervals is f(x) positive? Check all that apply.
(-infinity, -3)
(-3,-2)
(-2, 2)
(2,3)
(infinity, 3)

Answer 1

To determine for which intervals the function f(x) = (9 - x^2)/(x^2 - 4) is positive, we need to find the values of x that make the numerator positive and the denominator negative. The intervals (-3, -2) and (2, 3) are the intervals for which f(x) is positive.

Step-by-step explanation:

To determine for which intervals the function f(x) = (9 - x2)/(x2 - 4) is positive, we need to find the values of x that make the numerator positive and the denominator negative. Here are the steps to find the intervals:

1. Factor the numerator and denominator: (9 - x2) = (3 - x)(3 + x) and (x2 - 4) = (x - 2)(x + 2).
2. Find the critical points by setting each factor equal to zero and solving for x. For the numerator, x = 3 and x = -3. For the denominator, x = 2 and x = -2.
3. Create a number line with the critical points as intervals:

• Interval (-∞, -3): Test a value less than -3, like -4, in the function f(x) to see if it's positive or negative. f(-4) = (9 - (-4)^2) / ((-4)^2 - 4) = (9 - 16) / (16 - 4) = -7/12, which is negative.
• Interval (-3, -2): Test a value between -3 and -2, like -2.5. f(-2.5) = (9 - (-2.5)^2) / ((-2.5)^2 - 4) = (9 - 6.25) / (6.25 - 4) = 2.75 / 2.25, which is positive.
• Interval (-2, 2): Test a value between -2 and 2, like 0. f(0) = (9 - 0^2) / (0^2 - 4) = 9 / -4, which is negative.
• Interval (2, 3): Test a value between 2 and 3, like 2.5. f(2.5) = (9 - 2.5^2) / (2.5^2 - 4) = (9 - 6.25) / (6.25 - 4) = 2.75 / 2.25, which is positive.
• Interval (3, ∞): Test a value greater than 3, like 4. f(4) = (9 - 4^2) / (4^2 - 4) = (9 - 16) / (16 - 4) = -7/12, which is negative.

From the test values, we can see that f(x) is positive on the intervals (-3, -2) and (2, 3).