Answer :
Let's analyze the end behavior of the function [tex]\( f(x) = 5x^3 - 3x + 332 \)[/tex].
### Step 1: Degree and Leading Coefficient
The function [tex]\( f(x) \)[/tex] is a polynomial function. For polynomial functions, the end behavior is mainly determined by the term with the highest degree, which in this case is [tex]\( 5x^3 \)[/tex].
- The degree of the polynomial is [tex]\( 3 \)[/tex] (the exponent of [tex]\( x \)[/tex] in the term [tex]\( 5x^3 \)[/tex]).
- The leading coefficient is [tex]\( 5 \)[/tex], which is positive.
### Step 2: Behavior of Cubic Functions
For polynomials of the form [tex]\( ax^n \)[/tex]:
- If [tex]\( n \)[/tex] is odd and [tex]\( a > 0 \)[/tex] (like here where [tex]\( n = 3 \)[/tex] and [tex]\( a = 5 \)[/tex]), as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex] and as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
Thus, since the degree is 3 (odd) and the leading coefficient is positive (5), the right end of the graph will rise to positive infinity and the left end will fall to negative infinity.
### Step 3: Classification of the Function
For a function [tex]\( f(x) \)[/tex] to be identified as odd, it should satisfy [tex]\( f(-x) = -f(x) \)[/tex].
- Let's test [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = 5(-x)^3 - 3(-x) + 332 = -5x^3 + 3x + 332 \][/tex]
Notice that [tex]\( f(-x) \neq -f(x) \)[/tex]. Therefore, the function [tex]\( f(x) \)[/tex] is actually not odd.
This can be misleading at first since it certainly exhibits behavior that makes the ends go in opposite directions due to the odd power, but it's not an odd function in the strict mathematical sense.
### Conclusion
Given the leading coefficient is positive and the trend we determined:
- The function behavior at the right end (as [tex]\( x \to \infty \)[/tex]) is that [tex]\( f(x) \to \infty \)[/tex].
- The function behavior at the left end (as [tex]\( x \to -\infty \)[/tex]) is that [tex]\( f(x) \to -\infty \)[/tex].
Considering these observations, the correct option that correctly describes the end behavior:
Answer: B. The leading coefficient is positive so the left end goes down.
### Step 1: Degree and Leading Coefficient
The function [tex]\( f(x) \)[/tex] is a polynomial function. For polynomial functions, the end behavior is mainly determined by the term with the highest degree, which in this case is [tex]\( 5x^3 \)[/tex].
- The degree of the polynomial is [tex]\( 3 \)[/tex] (the exponent of [tex]\( x \)[/tex] in the term [tex]\( 5x^3 \)[/tex]).
- The leading coefficient is [tex]\( 5 \)[/tex], which is positive.
### Step 2: Behavior of Cubic Functions
For polynomials of the form [tex]\( ax^n \)[/tex]:
- If [tex]\( n \)[/tex] is odd and [tex]\( a > 0 \)[/tex] (like here where [tex]\( n = 3 \)[/tex] and [tex]\( a = 5 \)[/tex]), as [tex]\( x \to \infty \)[/tex], [tex]\( f(x) \to \infty \)[/tex] and as [tex]\( x \to -\infty \)[/tex], [tex]\( f(x) \to -\infty \)[/tex].
Thus, since the degree is 3 (odd) and the leading coefficient is positive (5), the right end of the graph will rise to positive infinity and the left end will fall to negative infinity.
### Step 3: Classification of the Function
For a function [tex]\( f(x) \)[/tex] to be identified as odd, it should satisfy [tex]\( f(-x) = -f(x) \)[/tex].
- Let's test [tex]\( f(-x) \)[/tex]:
[tex]\[ f(-x) = 5(-x)^3 - 3(-x) + 332 = -5x^3 + 3x + 332 \][/tex]
Notice that [tex]\( f(-x) \neq -f(x) \)[/tex]. Therefore, the function [tex]\( f(x) \)[/tex] is actually not odd.
This can be misleading at first since it certainly exhibits behavior that makes the ends go in opposite directions due to the odd power, but it's not an odd function in the strict mathematical sense.
### Conclusion
Given the leading coefficient is positive and the trend we determined:
- The function behavior at the right end (as [tex]\( x \to \infty \)[/tex]) is that [tex]\( f(x) \to \infty \)[/tex].
- The function behavior at the left end (as [tex]\( x \to -\infty \)[/tex]) is that [tex]\( f(x) \to -\infty \)[/tex].
Considering these observations, the correct option that correctly describes the end behavior:
Answer: B. The leading coefficient is positive so the left end goes down.
