Question

Determine the end behavior of the given polynomial function:

f(x) = 5x^4 + 3x^2 - 1

1. As x -> ∞ f(x) -> - ∞ and as x -> - ∞ f(x) -> ∞ .

2. As x -> ∞ f(x) -> ∞ and as x -> - ∞ f(x) -> - ∞ .

3. As x -> ∞ , f(x) -> ∞ and as x -> - ∞ , f(x) -> ∞ .

4. As x -> ∞ f(x) -> - ∞ and as x →-∞ f(x) -> - ∞

Which is the right answer?

Answer 1

3

Explanation:

When we look at the end behavior of a function, we mostly need to pay attention to its "dominant term," the one that grows quicker than the rest. Here, that term is 5x⁴, which explodes past the 3x² and -1 terms as x grows in the positive direction and in the negative direction. Note that 5x⁴ > 0 whether x > 0 or x < 0, so f(x) -> ∞ both as x -> ∞ and as x -> -∞

(If you need to be convinced of that last fact, remember than any negative number taken to an even power becomes positive. (-1)² = (-1)⁴ = (-1)⁶ = 1)

Answer 2

3. As x → ∞ , f(x) → ∞ and as x → -∞ , f(x) → ∞ .

Explanation:

When a number is raised to an even power, the result is always positive, regardless if the number is negative or positive.

Therefore, as the coefficients of the x⁴ and x² terms in the given polynomial function are both positive, then 5x⁴ ≥ 0 and 3x² ≥ 0.

As x becomes very large, the negative constant (-1) of the polynomial becomes negligible.

Therefore, as x approaches both positive and negative infinity, the function approaches positive infinity:

• As x → ∞ , f(x) → ∞.
• As x → -∞ , f(x) → ∞ .