Answer: The odd function is (D) [tex]f(x)=6x^3+2x.[/tex]
Step-by-step explanation: We are given to select the odd function from the options.
AN ODD FUNCTION: A function f(x) is said to be odd if f of negative x results in negative of f of x.
That is, f(-x) = - f(x).
Option (A) is
[tex]f(x)=3x^2+x.[/tex]
Putting x = -x, we have
[tex]f(-x)=3(-x)^2+(-x)=3x^2-x\neq -f(x).[/tex]
So, this function is not odd.
Option (B) is
[tex]f(x)=4x^3+7.[/tex]
Putting x = -x, we have
[tex]f(-x)=4(-x)^3+7=-4x^3+7\neq -f(x).[/tex]
So, this function is not odd.
Option (C) is
[tex]f(x)=5x^2+9.[/tex]
Putting x = -x, we have
[tex]f(-x)=5(-x)^2+9=5x^2+9\neq -f(x).[/tex]
So, this function is not odd.
Option (D) is
[tex]f(x)=6x^3+2x.[/tex]
Putting x = -x, we have
[tex]f(-x)=6(-x)^3+2(-x)=-6x^3-2x=-(6x^3+2x)=-f(x).[/tex]
So, this function is odd.
Therefore, the correct odd function is [tex]f(x)=6x^3+2x.[/tex]
Thus, (D) is the correct option.
