Answer:
a. f (x) < 0 for x ∈ (-∞ ,-3)
b. f (x) > 0 for x ∈ (-3,-2)
c. f (x) < 0 for x ∈ (-2,2)
d. f (x) > 0 for x ∈ (2,3)
e.f (x) < 0 for x ∈ (3,∞)
Step-by-step explanation:
Here, the given function is:![f(x)= \frac{9-x^2}{x^2-4}](https://tex.z-dn.net/?f=f%28x%29%3D%20%20%20%5Cfrac%7B9-x%5E2%7D%7Bx%5E2-4%7D)
Now, to check for the sign of f(x) at x = k, put the value of x from the given interval.
We get:
<u>a. (-infinity, -3)
</u>
put k = -4 from the given interval
We get ![f(-4)= \frac{9-(-4)^2}{(-4)^2-4} = \frac{9-16}{16-4} = \frac{-7}{12}](https://tex.z-dn.net/?f=f%28-4%29%3D%20%20%20%5Cfrac%7B9-%28-4%29%5E2%7D%7B%28-4%29%5E2-4%7D%20%20%3D%20%5Cfrac%7B9-16%7D%7B16-4%7D%20%20%3D%20%5Cfrac%7B-7%7D%7B12%7D%20%20%3C0)
⇒ f (x) < 0 for x ∈ (-∞ ,-3)
b. (-3, -2)
put k = -2.5 from the given interval
We get ![f(-2.5)= \frac{9-(-2.5)^2}{(-2.5)^2-4} = \frac{9-6.25}{6.25-4} = \frac{2.75}{2.25} > 0](https://tex.z-dn.net/?f=f%28-2.5%29%3D%20%20%20%5Cfrac%7B9-%28-2.5%29%5E2%7D%7B%28-2.5%29%5E2-4%7D%20%20%3D%20%5Cfrac%7B9-6.25%7D%7B6.25-4%7D%20%20%3D%20%5Cfrac%7B2.75%7D%7B2.25%7D%20%20%3E%200)
⇒ f (x) > 0 for x ∈ (-3,-2)
c. (-2, 2)
put k = 0 from the given interval
We get ![f(0)= \frac{9-(0)^2}{(0)^2-4} = \frac{9}{-4} = -\frac{9}{4} < 0](https://tex.z-dn.net/?f=f%280%29%3D%20%20%20%5Cfrac%7B9-%280%29%5E2%7D%7B%280%29%5E2-4%7D%20%20%3D%20%5Cfrac%7B9%7D%7B-4%7D%20%20%3D%20-%5Cfrac%7B9%7D%7B4%7D%20%20%3C%200)
⇒ f (x) < 0 for x ∈ (-2,2)
d. (2, 3)
put k =2.5 from the given interval
We get ![f(2.5)= \frac{9-(2.5)^2}{(2.5)^2-4} = \frac{9-6.25}{6.25-4} = \frac{2.75}{2.25} > 0](https://tex.z-dn.net/?f=f%282.5%29%3D%20%20%20%5Cfrac%7B9-%282.5%29%5E2%7D%7B%282.5%29%5E2-4%7D%20%20%3D%20%5Cfrac%7B9-6.25%7D%7B6.25-4%7D%20%20%3D%20%5Cfrac%7B2.75%7D%7B2.25%7D%20%20%3E%200)
⇒ f (x) > 0 for x ∈ (2,3)
e. (infinity, 3)
put k = 4 from the given interval
We get ![f(4)= \frac{9-(4)^2}{(4)^2-4} = \frac{9-16}{16-4} = \frac{-7}{12}](https://tex.z-dn.net/?f=f%284%29%3D%20%20%20%5Cfrac%7B9-%284%29%5E2%7D%7B%284%29%5E2-4%7D%20%20%3D%20%5Cfrac%7B9-16%7D%7B16-4%7D%20%20%3D%20%5Cfrac%7B-7%7D%7B12%7D%20%20%3C0)
⇒ f (x) < 0 for x ∈ (3,∞)