Question 5 of 10:

What can you say about the end behavior of the function [tex]$f(x)=5x^3-3x+332$[/tex]?

A. [tex]f(x)[/tex] is an odd function, so both ends go in the same direction.
B. [tex]f(x)[/tex] is an odd function, so both ends go in opposite directions.
C. The leading coefficient is positive, so the right end goes down.
D. The leading coefficient is positive, so the left end goes down.



Answer :

To analyze the end behavior of the function [tex]\( f(x) = 5x^3 - 3x + 332 \)[/tex], let's break down the components and properties of this polynomial.

1. Identify the leading term:
The function [tex]\( f(x) \)[/tex] is a polynomial, and the term with the highest power of [tex]\( x \)[/tex] dictates the end behavior of the function. Here, the leading term is [tex]\( 5x^3 \)[/tex].

2. Consider the leading coefficient:
In the term [tex]\( 5x^3 \)[/tex], the leading coefficient is 5, which is positive.

3. Examine the degree of the polynomial:
The degree of the polynomial is the highest power of [tex]\( x \)[/tex], which in this case is 3. Since 3 is odd, the end behavior will differ at [tex]\( x \to \infty \)[/tex] and [tex]\( x \to -\infty \)[/tex].

Based on these components:

- When [tex]\( x \to \infty \)[/tex] (as [tex]\( x \)[/tex] approaches positive infinity), the term [tex]\( 5x^3 \)[/tex] grows rapidly in the positive direction because the coefficient 5 is positive. Therefore, [tex]\( f(x) \)[/tex] will also tend to [tex]\( \infty \)[/tex].

- When [tex]\( x \to -\infty \)[/tex] (as [tex]\( x \)[/tex] approaches negative infinity), the term [tex]\( 5x^3 \)[/tex] grows rapidly in the negative direction (since [tex]\( x^3 \)[/tex] becomes more negative and is multiplied by 5). Therefore, [tex]\( f(x) \)[/tex] will tend to [tex]\( -\infty \)[/tex].

Hence, for an odd-degree polynomial with a positive leading coefficient, the left end (as [tex]\( x \to -\infty \)[/tex]) goes down (towards negative infinity) and the right end (as [tex]\( x \to \infty \)[/tex]) goes up (towards positive infinity).

This perfectly matches option:
D. The leading coefficient is positive so the left end goes down.

Therefore, the correct choice is [tex]\(\boxed{D}\)[/tex].