Question
If \( f(x)=x^{3}+\sin x \) then \( f(x) \) is: \( \begin{array}{ll}\text { (A) Even function } & \text { (B) Odd function } \\ \text { (C) Neither even nor odd function } & \text { (D) Constant function }\end{array} \)
Ask by Marshall Beck. in Pakistan
Feb 04,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( f(x) = x^3 + \sin x \) is an odd function.
Solution
To determine whether the function \( f(x) = x^3 + \sin x \) is even, odd, neither, or constant, let's analyze its symmetry properties.
**Definitions:**
- **Even Function:** A function \( f \) is even if \( f(-x) = f(x) \) for all \( x \).
- **Odd Function:** A function \( f \) is odd if \( f(-x) = -f(x) \) for all \( x \).
**Step-by-Step Analysis:**
1. **Compute \( f(-x) \):**
\[
f(-x) = (-x)^3 + \sin(-x) = -x^3 - \sin x
\]
2. **Compare \( f(-x) \) with \( f(x) \) and \( -f(x) \):**
- Original function: \( f(x) = x^3 + \sin x \)
- Negative of the function: \( -f(x) = -x^3 - \sin x \)
3. **Determine the Nature of \( f(x) \):**
\[
f(-x) = -x^3 - \sin x = -f(x)
\]
This satisfies the condition for an odd function.
**Conclusion:**
The function \( f(x) = x^3 + \sin x \) is an **odd function**.
**Answer:**
(B) Odd function
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To determine the nature of the function \( f(x) = x^3 + \sin x \), we can test if it's even, odd, or neither. A function is even if \( f(-x) = f(x) \) for all \( x \), and odd if \( f(-x) = -f(x) \). Calculating \( f(-x) \): \[ f(-x) = (-x)^3 + \sin(-x) = -x^3 - \sin x \] Now, check if it's odd: \[ -f(x) = - (x^3 + \sin x) = -x^3 - \sin x \] Since \( f(-x) = -f(x) \), we see that the function is odd. Hence, the correct answer is: (B) Odd function
