Question

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

Answer 1

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

• Option B: $y = -2x^3-1$
• Option C:  $y = -3x^3 + 4$

How to find the value of the function as x approaches infinity(+ve or -ve)?

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 ∞:

• Case A: $y = (x+ 2)^3 -9$

1. For x approaching -∞

$lim_{x \rightarrow -\infty} (y) = lim_{x \rightarrow -\infty} ((x+2)^3 - 9) = -9 + lim_{x \rightarrow -\infty} ((x+2)^3) \rightarrow - \infty$

(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.

• Case B: $y = -2x^3-1$

1. For x approaching -∞

$lim_{x \rightarrow -\infty} (y) = lim_{x \rightarrow -\infty} (-2x^3 - 1) = -1 + lim_{x \rightarrow -\infty} (-2x^3) \rightarrow \infty$

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

2.  For x approaching ∞

$lim_{x \rightarrow \infty} (y) = lim_{x \rightarrow \infty} (-2x^3 - 1) = -1 + lim_{x \rightarrow \infty} (-2x^3) \rightarrow -\infty$

So function approaches negative infinity as x approaches positive infinity

Thus, the considered function behaves as needed.

• Case C: $y = -3x^3 + 4$

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.

• Case D: $y = 3(x-1)^2$

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:

• Option B: $y = -2x^3-1$
• Option C:  $y = -3x^3 + 4$

Learn more about limiting values of a function here:

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