Question

the function f(x) is defined below. what is the end behavior of f(x)?

f(x) = 105x - x^5 - 104x^3 + 200 - 182x^2 - 18x^4

answer

as x → ∞, f(x) → ∞ and
as x → -∞, f(x) → ∞
as x → ∞, f(x) → -∞ and
as x → -∞, f(x) → ∞
as x → ∞, f(x) → -∞ and
as x → -∞, f(x) → -∞

Explanation:

Step1: Identify Leading Term

The leading term of a polynomial is the term with the highest degree. For \( f(x) = 105x - x^5 - 104x^3 + 200 - 182x^2 - 18x^4 \), the degrees of the terms are: \( 105x \) (degree 1), \( -x^5 \) (degree 5), \( -104x^3 \) (degree 3), \( 200 \) (degree 0), \( -182x^2 \) (degree 2), \( -18x^4 \) (degree 4). The highest degree is 5, so the leading term is \( -x^5 \).

Step2: Analyze End Behavior

For a polynomial with leading term \( ax^n \):

• If \( n \) is odd:
• If \( a > 0 \), as \( x \to \infty \), \( f(x) \to \infty \); as \( x \to -\infty \), \( f(x) \to -\infty \).
• If \( a < 0 \), as \( x \to \infty \), \( f(x) \to -\infty \); as \( x \to -\infty \), \( f(x) \to \infty \).

Here, \( a = -1 \) (coefficient of \( -x^5 \)) and \( n = 5 \) (odd). So:

• As \( x \to \infty \), \( -x^5 \to -\infty \) (since \( x^5 \) grows to \( \infty \) and we have a negative sign).
• As \( x \to -\infty \), \( -x^5 \to \infty \) (since \( x^5 \) is negative when \( x \) is negative, and multiplying by -1 makes it positive, so \( -(-\infty)^5 = -(-\infty) = \infty \)).

Answer:

as \( x \to \infty \), \( f(x) \to -\infty \) and as \( x \to -\infty \), \( f(x) \to \infty \) (the second option in the list)