Which of the following functions is an odd function:

Options

A. f(x)=constant
B. f(x)=sin x+cos x
C. `f(x)=sin(log (x+sqrt(x^(2)+1))`
D. `f(x)=1+x+2x^(3)`

Answer

To determine which of the given functions is an odd function, we need to check if they satisfy the condition for odd functions, which states that a function \( f(x) \) is odd if:

\[ f(-x) = -f(x) \]

Let's analyze each option step by step.

### Step 1: Analyze Option A
**Function:** \( f(x) = k \) (a constant)

1. Calculate \( f(-x) \):
\[ f(-x) = k \]
2. Calculate \( -f(x) \):
\[ -f(x) = -k \]
3. Compare \( f(-x) \) and \( -f(x) \):
\[ k \neq -k \quad \text{(unless \( k = 0 \))} \]
**Conclusion:** This function is not odd.

### Step 2: Analyze Option B
**Function:** \( f(x) = \sin x + \cos x \)

1. Calculate \( f(-x) \):
\[ f(-x) = \sin(-x) + \cos(-x) = -\sin x + \cos x \]
2. Calculate \( -f(x) \):
\[ -f(x) = -(\sin x + \cos x) = -\sin x - \cos x \]
3. Compare \( f(-x) \) and \( -f(x) \):
\[ -\sin x + \cos x \neq -\sin x - \cos x \]
**Conclusion:** This function is not odd.

### Step 3: Analyze Option C
**Function:** \( f(x) = \sin(\log x) + \sqrt{x^2 + 1} \)

1. Calculate \( f(-x) \):
\[ f(-x) = \sin(\log(-x)) + \sqrt{(-x)^2 + 1} = \sin(\log(-x)) + \sqrt{x^2 + 1} \]
Since \( \log(-x) = \log x + i\pi \) (not real for negative \( x \)), we will simplify this further.
2. We can express \( \sin(\log(-x)) \) as \( -\sin(\log x) \) (using the property of sine).
3. Therefore, we have:
\[ f(-x) = -\sin(\log x) + \sqrt{x^2 + 1} \]
4. Calculate \( -f(x) \):
\[ -f(x) = -(\sin(\log x) + \sqrt{x^2 + 1}) = -\sin(\log x) - \sqrt{x^2 + 1} \]
5. Compare \( f(-x) \) and \( -f(x) \):
\[ -\sin(\log x) + \sqrt{x^2 + 1} \neq -\sin(\log x) - \sqrt{x^2 + 1} \]
**Conclusion:** This function is not odd.

### Step 4: Analyze Option D
**Function:** \( f(x) = 1 + x + 2x^3 \)

1. Calculate \( f(-x) \):
\[ f(-x) = 1 - x - 2x^3 \]
2. Calculate \( -f(x) \):
\[ -f(x) = -(1 + x + 2x^3) = -1 - x - 2x^3 \]
3. Compare \( f(-x) \) and \( -f(x) \):
\[ 1 - x - 2x^3 \neq -1 - x - 2x^3 \]
**Conclusion:** This function is not odd.

### Final Conclusion
After analyzing all options, we find that none of the functions given are odd functions.