Which of the following function is an even function ?

Options

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

Answer

To determine which of the given functions is an even function, we will analyze each function using the definition of even and odd functions.

### Step-by-Step Solution:

1. **Understanding Even and Odd Functions**:
- An even function satisfies the condition: \( f(-x) = f(x) \) for all \( x \).
- An odd function satisfies the condition: \( f(-x) = -f(x) \) for all \( x \).

2. **Analyzing the First Function**:
- Given function: \( f(x) = \frac{a^x + a^{-x}}{a^x - a^{-x}} \)
- Calculate \( f(-x) \):
\[ f(-x) = \frac{a^{-x} + a^{x}}{a^{-x} - a^{x}} = \frac{a^{x} + a^{-x}}{-(a^{x} - a^{-x})} = -\frac{a^{x} + a^{-x}}{a^{x} - a^{-x}} = -f(x) \]
- Conclusion: This function is **odd**.

3. **Analyzing the Second Function**:
- Given function: \( f(x) = \frac{a^x + 1}{a^x - 1} \)
- Calculate \( f(-x) \):
\[ f(-x) = \frac{a^{-x} + 1}{a^{-x} - 1} = \frac{\frac{1}{a^x} + 1}{\frac{1}{a^x} - 1} = \frac{1 + a^x}{1 - a^x} \]
- Simplifying:
\[ f(-x) = \frac{1 + a^x}{1 - a^x} = -\frac{a^x + 1}{a^x - 1} = -f(x) \]
- Conclusion: This function is **odd**.

4. **Analyzing the Third Function**:
- Given function: \( f(x) = x \cdot \frac{a^x + 1}{a^x - 1} \)
- Calculate \( f(-x) \):
\[ f(-x) = -x \cdot \frac{a^{-x} + 1}{a^{-x} - 1} = -x \cdot \frac{1 + a^x}{1 - a^x} \]
- Simplifying:
\[ f(-x) = x \cdot \frac{a^x + 1}{a^x - 1} = f(x) \]
- Conclusion: This function is **even**.

5. **Analyzing the Fourth Function**:
- Given function: \( f(x) = \log_2(x + \sqrt{x^2 + 1}) \)
- Calculate \( f(-x) \):
\[ f(-x) = \log_2(-x + \sqrt{(-x)^2 + 1}) = \log_2(-x + \sqrt{x^2 + 1}) \]
- We can check if \( f(-x) + f(x) = 0 \):
\[ f(-x) + f(x) = \log_2(-x + \sqrt{x^2 + 1}) + \log_2(x + \sqrt{x^2 + 1}) = \log_2((-x + \sqrt{x^2 + 1})(x + \sqrt{x^2 + 1})) \]
- This simplifies to \( \log_2(1) = 0 \), thus \( f(-x) + f(x) = 0 \).
- Conclusion: This function is **odd**.

### Final Conclusion:
The only function that is an even function is the third function:
\[ f(x) = x \cdot \frac{a^x + 1}{a^x - 1} \]