To determine which of the given functions is an odd function, we need to check each function to see if it satisfies the condition for being an odd function. A function [tex]\( f(x) \)[/tex] is odd if for every [tex]\( x \)[/tex] in its domain, [tex]\( f(-x) = -f(x) \)[/tex].
### Checking [tex]\( f(x) = 3x^2 + x \)[/tex]
1. Compute [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 3(-x)^2 + (-x) = 3x^2 - x
\][/tex]
2. Compare [tex]\( f(-x) \)[/tex] with [tex]\(-f(x)\)[/tex]:
[tex]\[
-f(x) = -(3x^2 + x) = -3x^2 - x
\][/tex]
Clearly,
[tex]\[
f(-x) = 3x^2 - x \quad \text{and} \quad -f(x) = -3x^2 - x
\][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex], the function [tex]\( f(x) = 3x^2 + x \)[/tex] is not an odd function.
### Checking [tex]\( f(x) = 4x^3 + 7 \)[/tex]
1. Compute [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 4(-x)^3 + 7 = -4x^3 + 7
\][/tex]
2. Compare [tex]\( f(-x) \)[/tex] with [tex]\(-f(x)\)[/tex]:
[tex]\[
-f(x) = -(4x^3 + 7) = -4x^3 - 7
\][/tex]
Clearly,
[tex]\[
f(-x) = -4x^3 + 7 \quad \text{and} \quad -f(x) = -4x^3 - 7
\][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex], the function [tex]\( f(x) = 4x^3 + 7 \)[/tex] is not an odd function.
### Checking [tex]\( f(x) = 5x^2 + 9 \)[/tex]
1. Compute [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 5(-x)^2 + 9 = 5x^2 + 9
\][/tex]
2. Compare [tex]\( f(-x) \)[/tex] with [tex]\(-f(x)\)[/tex]:
[tex]\[
-f(x) = -(5x^2 + 9) = -5x^2 - 9
\][/tex]
Clearly,
[tex]\[
f(-x) = 5x^2 + 9 \quad \text{and} \quad -f(x) = -5x^2 - 9
\][/tex]
Since [tex]\( f(-x) \neq -f(x) \)[/tex], the function [tex]\( f(x) = 5x^2 + 9 \)[/tex] is not an odd function.
### Checking [tex]\( f(x) = 6x^3 + 2x \)[/tex]
1. Compute [tex]\( f(-x) \)[/tex]:
[tex]\[
f(-x) = 6(-x)^3 + 2(-x) = -6x^3 - 2x
\][/tex]
2. Compare [tex]\( f(-x) \)[/tex] with [tex]\(-f(x)\)[/tex]:
[tex]\[
-f(x) = -(6x^3 + 2x) = -6x^3 - 2x
\][/tex]
Clearly,
[tex]\[
f(-x) = -6x^3 - 2x \quad \text{and} \quad -f(x) = -6x^3 - 2x
\][/tex]
Since [tex]\( f(-x) = -f(x) \)[/tex], the function [tex]\( f(x) = 6x^3 + 2x \)[/tex] is an odd function.
Therefore, from the given functions, the only odd function is [tex]\( f(x) = 6x^3 + 2x \)[/tex].
