To determine which of the given functions is odd, we need to understand the definition of an odd function. A function [tex]\( f(x) \)[/tex] is said to be odd if for every [tex]\( x \)[/tex] in the function's domain, [tex]\( f(-x) = -f(x) \)[/tex].
Let's check each function to see if it satisfies this condition:
1. [tex]\( f(x) = 6x^3 + 2x \)[/tex]
We need to check if [tex]\( f(-x) = -f(x) \)[/tex]:
[tex]\[
f(-x) = 6(-x)^3 + 2(-x) = -6x^3 - 2x
\][/tex]
[tex]\[
-f(x) = -(6x^3 + 2x) = -6x^3 - 2x
\][/tex]
Since [tex]\( f(-x) = -f(x) \)[/tex], this function is odd.
2. [tex]\( f(x) = 3x^2 + x \)[/tex]
We need to check if [tex]\( f(-x) = -f(x) \)[/tex]:
[tex]\[
f(-x) = 3(-x)^2 + (-x) = 3x^2 - x
\][/tex]
[tex]\[
-f(x) = -(3x^2 + x) = -3x^2 - x
\][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex], this function is not odd.
3. [tex]\( f(x) = 4x^3 + 7 \)[/tex]
We need to check if [tex]\( f(-x) = -f(x) \)[/tex]:
[tex]\[
f(-x) = 4(-x)^3 + 7 = -4x^3 + 7
\][/tex]
[tex]\[
-f(x) = -(4x^3 + 7) = -4x^3 - 7
\][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex], this function is not odd.
4. [tex]\( f(x) = 5x^2 + 9 \)[/tex]
We need to check if [tex]\( f(-x) = -f(x) \)[/tex]:
[tex]\[
f(-x) = 5(-x)^2 + 9 = 5x^2 + 9
\][/tex]
[tex]\[
-f(x) = -(5x^2 + 9) = -5x^2 - 9
\][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex], this function is not odd.
From the above steps, we can conclude that the only odd function among the given options is [tex]\( f(x) = 6x^3 + 2x \)[/tex]. Therefore, the answer is:
[tex]\[ \boxed{1} \][/tex]
