To determine which of the given functions is an odd function, let's recall the definition of an odd function. A function [tex]\(f(x)\)[/tex] is considered odd if it satisfies the condition [tex]\( f(-x) = -f(x) \)[/tex] for all [tex]\( x \)[/tex] in its domain.
### Step-by-Step Analysis:
1. First Function: [tex]\( f(x) = 3x^2 + x \)[/tex]
[tex]\[
f(-x) = 3(-x)^2 + (-x) = 3x^2 - x
\][/tex]
Compare it with [tex]\( -f(x) \)[/tex]:
[tex]\[
-f(x) = -(3x^2 + x) = -3x^2 - x
\][/tex]
Since [tex]\( 3x^2 - x \neq -3x^2 - x \)[/tex], [tex]\( f(x) = 3x^2 + x \)[/tex] is not an odd function.
2. Second Function: [tex]\( f(x) = 4x^3 + 7 \)[/tex]
[tex]\[
f(-x) = 4(-x)^3 + 7 = -4x^3 + 7
\][/tex]
Compare it with [tex]\( -f(x) \)[/tex]:
[tex]\[
-f(x) = -(4x^3 + 7) = -4x^3 - 7
\][/tex]
Since [tex]\( -4x^3 + 7 \neq -4x^3 - 7 \)[/tex], [tex]\( f(x) = 4x^3 + 7 \)[/tex] is not an odd function.
3. Third Function: [tex]\( f(x) = 5x^2 + 9 \)[/tex]
[tex]\[
f(-x) = 5(-x)^2 + 9 = 5x^2 + 9
\][/tex]
Compare it with [tex]\( -f(x) \)[/tex]:
[tex]\[
-f(x) = -(5x^2 + 9) = -5x^2 - 9
\][/tex]
Since [tex]\( 5x^2 + 9 \neq -5x^2 - 9 \)[/tex], [tex]\( f(x) = 5x^2 + 9 \)[/tex] is not an odd function.
4. Fourth Function: [tex]\( f(x) = 6x^3 + 2x \)[/tex]
[tex]\[
f(-x) = 6(-x)^3 + 2(-x) = -6x^3 - 2x
\][/tex]
Compare it with [tex]\( -f(x) \)[/tex]:
[tex]\[
-f(x) = -(6x^3 + 2x) = -6x^3 - 2x
\][/tex]
Since [tex]\( -6x^3 - 2x = -6x^3 - 2x \)[/tex], [tex]\( f(x) = 6x^3 + 2x \)[/tex] is indeed an odd function.
### Conclusion:
Out of the given functions, the only one that satisfies the condition for being an odd function is [tex]\( f(x) = 6x^3 + 2x \)[/tex]. Therefore, the correct answer is:
[tex]\[
\boxed{f(x) = 6x^3 + 2x}
\][/tex]
