Question

The function \( f: (-\infty, \infty) \to (-\infty, 1) \), defined by \[ f(x) = \frac{2^x - 2^{-x}}{2^x + 2^{-x}}, \] is:

• A. Onto but not one-one
• B. Both one-one and onto
• C. Neither one-one nor onto
• D. One-one but not onto

Hint:

To determine if a function is one-to-one or onto, analyze its behavior over its entire domain and check if each value of the range is mapped from exactly one value in the domain (one-to-one) or if every value in the range is achieved (onto).

Answer 1

The Correct Option is A

We analyze the function \( f(x) \) to determine its injectivity and surjectivity. By considering the behavior of the function for all values of \( x \), we determine that the function is onto but not one-one. Final Answer: Onto but not one-one.