Which of the following statements is true?

**A.**

f(x)=sin⁡2(x)+cos⁡2(x)+x3−1*f*(*x*)=sin2(*x*)+cos2(*x*)+*x*3−1 is neither odd nor even.

**B.**

f(x)=e−x+7x3+x*f*(*x*)=*e*−*x*+7*x*3+*x* is an odd function.

**C.**

f(x)=∣x3∣+x*f*(*x*)=∣∣​*x*3∣∣​+*x* is an odd function.

**D.**

f(x)=sin⁡2(x)+2cos⁡(x)*f*(*x*)=sin2(*x*)+2cos(*x*) is an even function.

**E.**

f(x)=ln⁡x2+1*f*(*x*)=ln*x*2+1 is an odd function.

Let's analyze each of the given functions to determine which statement is true.

### Option A: \( f(x) = \sin^2(x) + \cos^2(x) + x^3 - 1 \)
- The function includes several components:
  - \( \sin^2(x) + \cos^2(x) \) simplifies to 1, since \( \sin^2(x) + \cos^2(x) = 1 \) for all \( x \).
  - \( x^3 \) is an odd function (\( f(-x) = -x^3 \)).
  - The constant term \( -1 \) does not change with \( x \) and thus is even.
  
  Substituting \( -x \) into \( f(x) \):
  \[
  f(-x) = \sin^2(-x) + \cos^2(-x) + (-x)^3 - 1 = 1 - x^3 - 1 = -x^3
  \]
  This does not equal \( f(x) \) nor \( -f(x) \). Therefore, **Option A is correct**: this function is neither odd nor even.

### Option B: \( f(x) = e^{-x} + 7x^3 + x \)
- Substituting \( -x \) into \( f(x) \):
  \[
  f(-x) = e^{-(-x)} + 7(-x)^3 + (-x) = e^x - 7x^3 - x
  \]
  This is not equal to \( -f(x) = -\left(e^{-x} + 7x^3 + x\right) \). Therefore, **Option B is incorrect**: the function is not odd.

### Option C: \( f(x) = |x^3| + x \)
- Since \( |x^3| = x^3 \) for \( x \geq 0 \) and \( |x^3| = -x^3 \) for \( x < 0 \), substituting \( -x \) gives:
  \[
  f(-x) = |-x^3| - x = |x^3| - x = x^3 - x
  \]
  This is not equal to \( -f(x) = -\left(|x^3| + x\right) = -x^3 - x \). Therefore, **Option C is incorrect**: the function is not odd.

### Option D: \( f(x) = \sin^2(x) + 2\cos(x) \)
- Substituting \( -x \) into \( f(x) \):
  \[
  f(-x) = \sin^2(-x) + 2\cos(-x) = \sin^2(x) + 2\cos(x)
  \]
  Since \( f(-x) = f(x) \), **Option D is correct**: this function is even.

### Option E: \( f(x) = \ln(x^2 + 1) \)
- Substituting \( -x \) into \( f(x) \):
  \[
  f(-x) = \ln((-x)^2 + 1) = \ln(x^2 + 1)
  \]
  Since \( f(-x) = f(x) \), **Option E is incorrect**: the function is even, not odd.

### Conclusion:
- **Option D**: \( f(x) = \sin^2(x) + 2\cos(x) \) is an even function. This is the correct statement.

Would you like more details or have any questions? Here are five related questions you might consider:

1. How do you distinguish between even and odd functions using symmetry?
2. What is the significance of even and odd functions in real-world applications?
3. Can a function be neither odd nor even?
4. How does the composition of functions affect their even or odd nature?
5. How do logarithmic and exponential functions behave in terms of oddness or evenness?

**Tip:** To check if a function is even, see if substituting \( x \) with \( -x \) results in the same function. For odd functions, check if the result is the negative of the original function.

Explore the properties of even and odd functions through detailed analysis of mathematical statements. Learn how to determine if a function is even, odd, or neither with examples and explanations.

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