Question

section 4.1

1. what is the end behavior of the function $f(x) = x^7 + x^5 - x^3$?
2. state the left and right end behaviors of the graph of $f(x) = -x^3 + x^2 - 7$

Explanation:

Problem 1: End Behavior of \( f(x) = x^7 + x^5 - x^3 \)

Step 1: Identify Leading Term

The leading term of a polynomial is the term with the highest degree. For \( f(x) = x^7 + x^5 - x^3 \), the leading term is \( x^7 \). The degree of the leading term is 7 (odd), and the leading coefficient is 1 (positive).

Step 2: Apply End Behavior Rules

For polynomials:

• If the degree is odd and the leading coefficient is positive:
• As \( x \to +\infty \) (right end), \( f(x) \to +\infty \) (because \( x^{\text{odd}} \) grows to \( +\infty \) when \( x \) is large positive and coefficient is positive).
• As \( x \to -\infty \) (left end), \( f(x) \to -\infty \) (because \( x^{\text{odd}} \) becomes \( -\infty \) when \( x \) is large negative and coefficient is positive: \( (-\infty)^7 = -\infty \), times positive 1 is \( -\infty \)).

Step 1: Identify Leading Term

The leading term is \( -x^3 \). Degree is 3 (odd), leading coefficient is \( -1 \) (negative).

Step 2: Apply End Behavior Rules

• For odd degree and negative leading coefficient:
• As \( x \to +\infty \) (right end): \( x^3 \to +\infty \), times \( -1 \) gives \( f(x) \to -\infty \).
• As \( x \to -\infty \) (left end): \( x^3 \to -\infty \) (since \( (-\infty)^3 = -\infty \)), times \( -1 \) gives \( f(x) \to +\infty \) (because \( -(-\infty) = +\infty \)).

Answer:

(for Problem 1):
As \( x \to +\infty \), \( f(x) \to +\infty \); as \( x \to -\infty \), \( f(x) \to -\infty \)

Problem 2: End Behavior of \( f(x) = -x^3 + x^2 - 7 \)