Question

Which of the following is an odd function?

Answer 1

The right answer is the very last one though the second one also contains x^3 but it's already shifted because +7
The others, by the way, are even functions :)

Answer 2

Answer:  The odd function is (D) $f(x)=6x^3+2x.$

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

$f(x)=3x^2+x.$

Putting x = -x, we have

$f(-x)=3(-x)^2+(-x)=3x^2-x\neq -f(x).$

So, this function is not odd.

Option (B) is

$f(x)=4x^3+7.$

Putting x = -x, we have

$f(-x)=4(-x)^3+7=-4x^3+7\neq -f(x).$

So, this function is not odd.

Option (C) is

$f(x)=5x^2+9.$

Putting x = -x, we have

$f(-x)=5(-x)^2+9=5x^2+9\neq -f(x).$

So, this function is not odd.

Option (D) is

$f(x)=6x^3+2x.$

Putting x = -x, we have

$f(-x)=6(-x)^3+2(-x)=-6x^3-2x=-(6x^3+2x)=-f(x).$

So, this function is odd.

Therefore, the correct odd function is $f(x)=6x^3+2x.$

Thus, (D) is the correct option.