Question

describe the end behavior for the following function.
$f(x) = -x^5 + 3x^3 + 3$
select one:
○ a. falls to the left. rises to the right.
○ b. falls to the left. falls to the right.
○ c. rises to the left. falls to the right.
○ d. rises to the left. rises to the right.

Explanation:

Step1: Identify Leading Term

The leading term of \( f(x) = -x^5 + 3x^3 + 3 \) is \( -x^5 \). The degree of the leading term is 5 (odd) and the leading coefficient is -1 (negative).

Step2: Analyze End Behavior Rules

For a polynomial function, if the degree is odd:

• If the leading coefficient is positive: as \( x \to +\infty \), \( f(x) \to +\infty \) (rises to the right); as \( x \to -\infty \), \( f(x) \to -\infty \) (falls to the left).
• If the leading coefficient is negative: as \( x \to +\infty \), \( f(x) \to -\infty \) (falls to the right); as \( x \to -\infty \), \( f(x) \to +\infty \) (rises to the left).

Step3: Apply to Given Function

Here, degree is odd (5) and leading coefficient is negative (-1). So:

• As \( x \to +\infty \) (right end), \( f(x) \to -\infty \) (falls to the right).
• As \( x \to -\infty \) (left end), \( f(x) \to +\infty \) (rises to the left).

Answer:

c. Rises to the Left. Falls to the Right.