To determine which of the following functions is an odd function, let's first recall the definition of an odd function. A function $f(x)$ is called odd if it satisfies the condition:

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

Let's analyze each function given in the options:

Option A: $f(x) = \cos^3(x)$

• Since cosine is an even function ($\cos(-x) = \cos(x)$), raising it to any odd power, such as $3$, will also result in an even function. Therefore, $f(-x) = \cos^3(-x) = \cos^3(x)$, which implies $f(x)$ is not an odd function.

Option B: $f(x) = 7x^5 - x^3 + 1$

• Let's substitute $-x$ into the function: $f(-x) = 7(-x)^5 - (-x)^3 + 1 = -7x^5 + x^3 + 1$ This does not equal $-f(x)$ because of the constant term $+1$. Therefore, $f(x)$ is not an odd function.

Option C: $f(x) = (4x + 2x^{11})x^2$ and $f(x) = 2\sin(x)\cos(x)$

• For $f(x) = (4x + 2x^{11})x^2$: $f(-x) = (4(-x) + 2(-x)^{11})(-x)^2 = (-4x - 2x^{11})x^2 = -f(x)$ However, since it involves a product with an even power of $x^2$, $f(x)$ cannot be odd.

• For $f(x) = 2\sin(x)\cos(x)$: $f(-x) = 2\sin(-x)\cos(-x) = 2(-\sin(x))\cos(x) = -2\sin(x)\cos(x) = -f(x)$ So, this function is an odd function.

Option D: $f(x) = (4x + 2x^{11})x^2$

• This is the same as the first part of Option C. As discussed, $f(x)$ cannot be odd because of the $x^2$ term.

Option E: $f(x) = 2\sin(x)\cos(x)$

• As shown earlier, this function is odd.

Conclusion:

Option E: $f(x) = 2\sin(x)\cos(x)$ is an odd function.

Would you like to go into more detail or have any questions? Here are five related questions you might be interested in:

1. How do you determine if a function is even or odd?
2. What are some examples of even functions?
3. Can a function be both even and odd?
4. How does the symmetry of a graph relate to even and odd functions?
5. How does the addition or multiplication of functions affect their odd or even nature?

Tip: When determining if a function is odd, always remember to check the behavior of the function under the substitution $x$ with $-x$.