Which of the following functions is even,

Options

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

Answer

To determine which of the given functions is even, we need to recall the definition of an even function. A function \( f(x) \) is considered even if:

\[ f(-x) = f(x) \quad \text{for all } x \]

Let's analyze each of the options provided to see which one satisfies this condition.

### Step 1: Analyze the first function
Let’s denote the first function as \( f_1(x) \).

1. **Substituting**: Replace \( x \) with \( -x \):
\[ f_1(-x) = -(-x) = x \]
This does not equal \( f_1(x) \), so \( f_1(x) \) is not an even function.