Which of the given functions is (are) even, odd, and which of them is (are) neither even, nor odd?
`f(x)=log x+sqrt(1+x^(2))`, (b) `f(x)=log ""(1-x)/(1+x)`
(c) `f(x)=2x^(3)-x+1`, (d) `f(x)=x""(a^(x)+1)/(a^(x)-1)`

Answer

To determine whether the given functions are even, odd, or neither, we will analyze each function step by step.

### Given Functions:
1. \( f(x) = \log x + \sqrt{1 + x^2} \)
2. \( f(x) = \log \left( \frac{1-x}{1+x} \right) \)
3. \( f(x) = 2x^3 - x + 1 \)
4. \( f(x) = \frac{x(a^x + 1)}{a^x - 1} \)

### Step-by-Step Analysis:

#### Function (a): \( f(x) = \log x + \sqrt{1 + x^2} \)

1. **Calculate \( f(-x) \)**:
\[ f(-x) = \log(-x) + \sqrt{1 + (-x)^2} \]
Since \( \log(-x) \) is undefined for real \( x \), we cannot evaluate \( f(-x) \).

2. **Conclusion**:
Since \( f(-x) \) is undefined, \( f(x) \) is **neither even nor odd**.

#### Function (b): \( f(x) = \log \left( \frac{1-x}{1+x} \right) \)

1. **Calculate \( f(-x) \)**:
\[ f(-x) = \log \left( \frac{1 - (-x)}{1 + (-x)} \right) = \log \left( \frac{1 + x}{1 - x} \right) \]

2. **Check if \( f(-x) = -f(x) \)**:
\[ -f(x) = -\log \left( \frac{1-x}{1+x} \right) = \log \left( \frac{1+x}{1-x} \right) \]
Thus, \( f(-x) = -f(x) \).

3. **Conclusion**:
\( f(x) \) is an **odd function**.

#### Function (c): \( f(x) = 2x^3 - x + 1 \)

1. **Calculate \( f(-x) \)**:
\[ f(-x) = 2(-x)^3 - (-x) + 1 = -2x^3 + x + 1 \]

2. **Check if \( f(-x) = -f(x) \)**:
\[ -f(x) = -(2x^3 - x + 1) = -2x^3 + x - 1 \]
Since \( f(-x) \neq -f(x) \) and \( f(-x) \neq f(x) \), it is neither even nor odd.

3. **Conclusion**:
\( f(x) \) is **neither even nor odd**.

#### Function (d): \( f(x) = \frac{x(a^x + 1)}{a^x - 1} \)

1. **Calculate \( f(-x) \)**:
\[ f(-x) = \frac{-x(a^{-x} + 1)}{a^{-x} - 1} = \frac{-x \left( \frac{1}{a^x} + 1 \right)}{\frac{1}{a^x} - 1} \]
Simplifying gives:
\[ f(-x) = \frac{-x(1 + a^x)}{1 - a^x} \]

2. **Check if \( f(-x) = f(x) \)**:
We can see that:
\[ f(-x) = -f(x) \]
Thus, \( f(-x) = f(x) \).

3. **Conclusion**:
\( f(x) \) is an **even function**.

### Summary of Results:
- (a) **Neither even nor odd**
- (b) **Odd**
- (c) **Neither even nor odd**
- (d) **Even**