Question

consider the polynomial function $f(x) = -9x - 6x^2 + x^7 + 5x^9$. as $x \to -\infty$, $f(x) \to ?$ as $x \to \infty$, $f(x) \to ?$

Explanation:

Step 1: Identify the leading term

The leading term of a polynomial is the term with the highest degree. For the polynomial \( f(x) = -9x - 6x^2 + x^7 + 5x^9 \), the leading term is \( 5x^9 \) since the degree of \( x^9 \) (which is 9) is the highest among all the terms.

Step 2: Analyze the leading term as \( x \to -\infty \)

For a power function \( ax^n \), the end - behavior is determined by the sign of the leading coefficient \( a \) and the parity (even or odd) of the exponent \( n \).

• If \( n \) is odd:
• If \( a>0 \), as \( x\to -\infty \), \( ax^n\to -\infty \) (because when you raise a negative number to an odd power, the result is negative, and multiplying by a positive \( a \) still keeps it negative for large negative \( x \)).
• If \( a < 0 \), as \( x\to -\infty \), \( ax^n\to+\infty \) (because raising a negative number to an odd power gives a negative number, and multiplying by a negative \( a \) gives a positive number for large negative \( x \)).

For the leading term \( 5x^9 \), \( a = 5>0 \) and \( n=9 \) (which is odd). So when \( x\to -\infty \), \( 5x^9\to -\infty \). Since the leading term dominates the end - behavior of the polynomial, as \( x\to -\infty \), \( f(x)\to -\infty \).

Step 3: Analyze the leading term as \( x\to\infty \)

For a power function \( ax^n \) with \( n \) odd:

• If \( a>0 \), as \( x\to\infty \), \( ax^n\to\infty \) (because raising a positive number to an odd power gives a positive number, and multiplying by a positive \( a \) makes it large positive for large positive \( x \)).
• If \( a < 0 \), as \( x\to\infty \), \( ax^n\to -\infty \) (because raising a positive number to an odd power gives a positive number, and multiplying by a negative \( a \) gives a negative number for large positive \( x \)).

For the leading term \( 5x^9 \), \( a = 5>0 \) and \( n = 9 \) (odd). So when \( x\to\infty \), \( 5x^9\to\infty \). Since the leading term dominates the end - behavior of the polynomial, as \( x\to\infty \), \( f(x)\to\infty \).

Answer:

s:
As \( x\to -\infty \), \( f(x)\to\boldsymbol{-\infty} \)

As \( x\to\infty \), \( f(x)\to\boldsymbol{\infty} \)