Question

1. for a polynomial function with the form $f(x) = x^n$ where $n$ is odd, what is the end behavior?

a. rises to positive infinity at both ends
b. rises to positive infinity on the right and falls to negative infinity on the left
c. falls to negative infinity at both ends
d. falls to negative infinity on the right and rises on the left

Explanation:

Step1: Recall end behavior of \( f(x)=x^n \)

For a power function \( f(x) = x^n \), when \( n \) is odd:

• As \( x \to +\infty \), \( x^n \to +\infty \) (since positive number to odd power is positive, and magnitude increases).
• As \( x \to -\infty \), \( x^n \to -\infty \) (since negative number to odd power is negative, and magnitude increases).

Step2: Analyze options

• Option a: Rises at both ends. But for odd \( n \), left end ( \( x\to -\infty \)) goes to \( -\infty \) (falls), so a is wrong.
• Option b: Right end (\( x\to +\infty \)) rises (to \( +\infty \)), left end (\( x\to -\infty \)) falls (to \( -\infty \)). This matches the end behavior of \( f(x)=x^n \) with odd \( n \).
• Option c: Falls at both ends. Right end should rise, so c is wrong.
• Option d: Right end falls (wrong, should rise) and left end rises (wrong, should fall). So d is wrong.

Answer:

b. Rises to positive infinity on the right and falls to negative infinity on the left