Question

If \( f : \mathbb{R} \rightarrow \mathbb{R} \) such that \[ f(x) = \frac{e^{x} + e^{-x}}{e^{x} - e^{-x}}, \] then \( f \) is

• A. a periodic function
• B. an even function
• C. an odd function
• D. neither even nor odd function

Hint:

To test whether a function is even or odd, always compute \( f(-x) \) and compare it with \( f(x) \).

Answer 1

The Correct Option is C

Step 1: Replace \( x \) by \( -x \) in \( f(x) \).
\[ f(-x) = \frac{e^{-x} + e^{x}}{e^{-x} - e^{x}} \]

Step 2: Simplify the expression.
\[ f(-x) = \frac{e^{x} + e^{-x}}{-(e^{x} - e^{-x})} = -\frac{e^{x} + e^{-x}}{e^{x} - e^{-x}} \]

Step 3: Compare \( f(-x) \) with \( f(x) \).
\[ f(-x) = -f(x) \]

Step 4: Conclusion.
Since \( f(-x) = -f(x) \), the function is an odd function.