Which of the following function is even function?

Options

A. f(x)=log`(1-x)/(1+x)`
B. f(x)=`x.(a^x+1)/(a^x-1)`
C. f(x)=`log[x+sqrt(1+x^2)]`
D. f(x)=`sqrt(1+x+x^2)-sqrt(1-x+x^2)`

Answer

To determine which of the given functions is an even function, we need to check each function against the definition of an even function. A function \( f(x) \) is considered even if \( f(-x) = f(x) \) for all \( x \) in the domain of the function.

Let's analyze each function step by step:

### Step 1: Analyze the first function
**Function:** \( f(x) = \log\left(\frac{1-x}{1+x}\right) \)

**Check for evenness:**
1. Calculate \( f(-x) \):
\[ f(-x) = \log\left(\frac{1-(-x)}{1+(-x)}\right) = \log\left(\frac{1+x}{1-x}\right) \]
2. Notice that:
\[ f(-x) = \log\left(\frac{1+x}{1-x}\right) = -\log\left(\frac{1-x}{1+x}\right) = -f(x) \]
Since \( f(-x) = -f(x) \), this function is **odd**.

### Step 2: Analyze the second function
**Function:** \( f(x) = \frac{x \cdot 1^x + 1}{1^x - 1} \)

**Check for evenness:**
1. Calculate \( f(-x) \):
\[ f(-x) = \frac{-x \cdot 1^{-x} + 1}{1^{-x} - 1} \]
Since \( 1^{-x} = 1 \), we simplify:
\[ f(-x) = \frac{-x + 1}{1 - 1} \text{ (undefined)} \]
This function is not defined for \( x = 0 \) and does not yield a valid comparison, so we cannot conclude if it is even or odd.

### Step 3: Analyze the third function
**Function:** \( f(x) = \log(x + \sqrt{1+x^2}) \)

**Check for evenness:**
1. Calculate \( f(-x) \):
\[ f(-x) = \log(-x + \sqrt{1+(-x)^2}) = \log(-x + \sqrt{1+x^2}) \]
This does not simplify to \( f(x) \), and since \( \log(-x + \sqrt{1+x^2}) \) does not equal \( f(x) \), this function is neither even nor odd.

### Step 4: Analyze the fourth function
**Function:** \( f(x) = \sqrt{1+x} + \sqrt{1-x} \)

**Check for evenness:**
1. Calculate \( f(-x) \):
\[ f(-x) = \sqrt{1+(-x)} + \sqrt{1-(-x)} = \sqrt{1-x} + \sqrt{1+x} \]
Notice that:
\[ f(-x) = \sqrt{1-x} + \sqrt{1+x} = f(x) \]
Since \( f(-x) = f(x) \), this function is **even**.

### Conclusion
The only even function among the given options is the fourth function.

### Final Answer:
**The even function is:** \( f(x) = \sqrt{1+x} + \sqrt{1-x} \)