To determine which of the following functions is an odd function, we need to recall the definition of an odd function. A function $f(x)$ is odd if it satisfies the condition:

$f(-x) = -f(x)$

Let's check each function:

a) $y = 2x^3 - 3x^2$

Substitute $-x$ into the function:

$f(-x) = 2(-x)^3 - 3(-x)^2 = -2x^3 - 3x^2$

The expression $-2x^3 - 3x^2$ is not equal to $-(2x^3 - 3x^2)$, so this function is not odd.

b) $y = -2x^5 - 2x$

Substitute $-x$ into the function:

$f(-x) = -2(-x)^5 - 2(-x) = 2x^5 + 2x$

We get $f(-x) = 2x^5 + 2x$, and this is equal to $-( -2x^5 - 2x ) = 2x^5 + 2x$, so this function satisfies the condition for an odd function. Therefore, this is an odd function.

c) $y = 2x^3 - x + 1$

Substitute $-x$ into the function:

$f(-x) = 2(-x)^3 - (-x) + 1 = -2x^3 + x + 1$

The expression $-2x^3 + x + 1$ is not equal to $-(2x^3 - x + 1)$, so this function is not odd.

d) None of these

Since option b is an odd function, option d is not correct.

Conclusion:

The correct answer is b) $y = -2x^5 - 2x$.

Would you like any further details or clarification?

Related Questions:

1. How do you check if a function is even or odd?
2. What are the characteristics of even functions?
3. Can a function be both even and odd?
4. What is the geometric interpretation of an odd function?
5. How can symmetry help in identifying even or odd functions?

Tip:

Odd functions are symmetric about the origin, while even functions are symmetric about the y-axis.