Which function displays this end behavior?
• As xapproaches negative infinity, y approaches positive infinity.
• As xapproaches positive infinity, yapproaches negative infinity.
A.) y = (x+ 2)^3 -9
B.) y = -2x^3-1
C.) y = -3x^3 + 4
D.) y = 3(x-1)^2

The functions that displays the behavior that as x approaches -∞, y approaches ∞ and as x approaches -∞, y approaches  -∞ are:

If limits exist, we can take limits of the function, where x tends to -∞ or ∞, and that limiting value will be the value the function will approach.

Testing all the four functions listed for x tending to -∞ or ∞:

1. For x approaching -∞

[tex]lim_{x \rightarrow -\infty} (y) = lim_{x \rightarrow -\infty} ((x+2)^3 - 9) = -9 + lim_{x \rightarrow -\infty} ((x+2)^3) \rightarrow - \infty[/tex]

(since cube of negative quantity is going to be negative)

Thus, function also tends to -∞. So this function doesn't satisfy first case, thus, we don't need to test other case as only that function is needed which can satisfy both the cases.

1. For x approaching -∞

[tex]lim_{x \rightarrow -\infty} (y) = lim_{x \rightarrow -\infty} (-2x^3 - 1) = -1 + lim_{x \rightarrow -\infty} (-2x^3) \rightarrow \infty[/tex]

(this time, negative sign was canceled by outer negative. So function approaches positive infinity as x approaches negative infinity.

2.  For x approaching ∞

[tex]lim_{x \rightarrow \infty} (y) = lim_{x \rightarrow \infty} (-2x^3 - 1) = -1 + lim_{x \rightarrow \infty} (-2x^3) \rightarrow -\infty[/tex]

So function approaches negative infinity as x approaches positive infinity

Thus, the considered function behaves as needed.

Similar to above cases, we can see that negative sign and cube power will make this function go to positive infinity as x approaches negative infinity, and as x approaches positive infinity, the function will approach negative infinity.

Remember that finite constants added or subtracted from arbitrary large quantities(positive or negative infinity) can't affect them.

Here x is in square power, no matter if x approaches +ve or -ve infinity, the square term would make it positive. So this function cannot approach negative infinity.

Thus, only Option B and C are performing as needed. Therefore, the functions that displays the behavior that as x approaches -∞, y approaches ∞ and as x approaches -∞, y approaches  -∞ are:

Learn more about limiting values of a function here:

https://brainly.com/question/13146225

Answer: y=-2x^-1

Step-by-step explanation:

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