Question

what is the end behavior of the function $f(x) = 8x^5 - 3x^3 + 8x + 5$?
a. as $x$ approaches infinity, $f(x)$ approaches negative infinity, and as $x$ approaches negative infinity, $f(x)$ approaches negative infinity
b. as $x$ approaches infinity, $f(x)$ approaches infinity, and as $x$ approaches negative infinity, $f(x)$ approaches negative infinity
c. as $x$ approaches negative infinity, $f(x)$ approaches infinity, and as $x$ approaches infinity, $f(x)$ approaches negative infinity
d. as $x$ approaches infinity, $f(x)$ approaches negative infinity, and as $x$ approaches negative infinity, $f(x)$ approaches infinity

Explanation:

Step1: Identify the leading term

The function is \( f(x) = 8x^5 - 3x^3 + 8x + 5 \). The leading term (the term with the highest degree) is \( 8x^5 \).

Step2: Analyze the degree and leading coefficient

The degree of the leading term is 5 (which is odd), and the leading coefficient is 8 (which is positive).

Step3: Determine end behavior for odd degree with positive leading coefficient

For a polynomial function with an odd - degree leading term and a positive leading coefficient:

• As \( x\to\infty \), \( x^n\to\infty \) when \( n \) is odd and the coefficient is positive. So, as \( x\to\infty \), \( 8x^5\to\infty \), and since the other terms become negligible compared to the leading term as \( x \) approaches infinity, \( f(x)\to\infty \).
• As \( x\to-\infty \), \( x^n\to-\infty \) when \( n \) is odd (because a negative number raised to an odd power is negative) and the coefficient is positive. So, \( 8x^5\to-\infty \) as \( x\to-\infty \), and again, the other terms are negligible, so \( f(x)\to-\infty \) as \( x\to-\infty \).

Answer:

B. As \( x \) approaches infinity, \( f(x) \) approaches infinity, and as \( x \) approaches negative infinity, \( f(x) \) approaches negative infinity