Question

Can someone help me with this math homework please!

Answer 1

Answer:  Choice B

$\text{As } x \to \infty, \ f(x) \to \infty$  and $\text{As } x \to -\infty, \ f(x) \to \infty$

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Explanation:

Assuming that the only roots are x = -3 and x = 2, as shown by the table, then notice how f(x) > 0 at the very right hand side of the table. This indicates that f(x) will remain positive for this tail end. If f(x) were to become negative, then there would have to be another root somewhere beyond x = 2 (but that contradicts the assumption made at the top of the paragraph).

This allows us to say $\text{As } x \to \infty, \ f(x) \to \infty$

In other words, as x gets bigger, so does y. Both head to positive infinity together. We can informally describe this end behavior as "rises to the right".

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For the left side of the table, we also have positive f(x) values to the left of the root. So we see that f(x) > 0 as x heads off in the negative infinity direction.

We would then say $\text{As } x \to -\infty, \ f(x) \to \infty$

Regardless of what direction x goes (positive or negative infinity), the y = f(x) values are heading off to positive infinity. Informally, we could say "rises to the left".

Both endpoints rise together to positive infinity. Think of a parabolic curve that opens upward as one example.

All of this points to choice B as our final answer. Choice B can be rephrased as $\text{As } x \to \infty \text{ or } x \to -\infty, \text{ then } \ f(x) \to \infty$