Which of the following functions is an odd function? A. 𝑓 ( 𝑥 ) = cos ⁡ 3 ( 𝑥 ) f(x)=cos 3 (x) B. 𝑓 ( 𝑥 ) = 7 𝑥 5 − 𝑥 3 + 1 f(x)=7x 5 −x 3 +1 C. Both 𝑓 ( 𝑥 ) = ( 4 𝑥 + 2 𝑥 1 1 ) 𝑥 2 f(x)=(4x+2x 11 )x 2 and 𝑓 ( 𝑥 ) = 2 sin ⁡ ( 𝑥 ) cos ⁡ ( 𝑥 ) f(x)=2sin(x)cos(x). D. 𝑓 ( 𝑥 ) = ( 4 𝑥 + 2 𝑥 1 1 ) 𝑥 2 f(x)=(4x+2x 11 )x 2 E. 𝑓 ( 𝑥 ) = 2 sin ⁡ ( 𝑥 ) cos ⁡ ( 𝑥 ) f(x)=2sin(x)cos(x)

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 \( -x \).

Explore how to determine if a function is odd with detailed examples. Learn about cosine functions, polynomial functions, and trigonometric identities. Suitable for high school students studying functions in mathematics.

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