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In the function f(x) = a(x + 2)(x-3)^b, a and b are both
integer constants and b is positive. If the end behavior
of the graph of y = f(x) is positive for both very large
negative values of x and very large positive values of x,
what is true about a and b?
A. a is negative, and b is even.
B. a is positive, and b is even.
C. a is negative, and b is odd.
D. a is positive, and b is odd.

Sagot :

Using limits, considering the end behavior of the function f(x), it is found that the correct statement is given as follows:

D. a is positive, and b is odd.

What is the end behavior of a function?

It is given by it's limits as x goes to negative and positive infinity.

In this problem, we have that, considering the end behavior:

  • [tex]\lim_{x \rightarrow \infty} = \infty \rightarrow ax^{b + 1} = \infty[/tex]
  • [tex]\lim_{x \rightarrow -\infty} = \infty \rightarrow ax^{b + 1} = \infty[/tex]

This will only be true if a is positive and b is odd, hence option D is correct.

More can be learned about the end behavior of a function at https://brainly.com/question/27851082

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