Which statement is true about the discontinuities of the function f(x)?
f(x)=(x^2-4)/(x^3-x^2-2x)
a. there is a hole at x = 2.
b. there are asymptotes at x = 0 and x = –1.
c. there are asymptotes at x = 0 and x = –1 and a hole at (2, 2/3)
d. there are holes at x = 0 and x = –1 and an asymptote at x = 2.

Here we need to see which statements are true about the discontinuities of the given function.

The correct option is b: "There are asymptotes at x = 0 and x = –1."

So a rational function has discontinuities when its denominator becomes equal to zero, this is because we can't divide by zero.

Here the function is:

First, we should factorize both numerator and denominator, so we can see if some of the factors appear in both of them and we can remove it.

For the numerator we have:

For the denominator we have:

Now we need to find the factorization of the thing inside the parenthesis.

The roots are given by Bhaskara's formula:

So the roots are:

x = (1 + 3)/2 = 2

x = (1 - 3)/2 = -1

And we can rewrite:

Then the function can be rewritten as:

Then the discontinuities are at the zeros of the denominator, which are at x = 0 and x = -1.

Because the denominator goes to zero at these points, we will have vertical asymptotes, thus the correct option is b

"There are asymptotes at x = 0 and x = –1."

If you want to learn more, you can read:

link

the population of other specie of birds will decline or other specie of birds will find some alternate source of food to survive.

explanation:

according to the competitive exclusion principle, which is also known as gause's law, two specie living in the same niche and feeding on the same resource cannot co-exist stably.

as we have well studied the theory of natural selection which states that only those organisms survive which have better tendency to compete for food and adapt to the environmental conditions.

coming towards the question, what wi ll happen when one species outcompetes the other for the small fish , the other specie will not get enough food and will start to die or decline in population. eventually either other specie needs to find another alternative food source for itself or migrate at some other location where there is less competition otherwise it will just not exist anymore at that niche.

hope it !

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step-by-step explanation: