Let \(f : (0, \infty) \to R\) be defined by \(f(x) = \log_e x\), where R is the set of real numbers. Then f is
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → one-one and onto
One-one: if \(f(p) = f(q)\) for positive p and q, then
\(\log_e p = \log_e q\)
\(e^{\log_e p} = e^{\log_e q}\)
\(p = q\)
Onto: for any real y, the number \(x = e^{y}\) is positive, so it lies in \((0, \infty)\), and
\(f(e^{y}) = \log_e e^{y} = y\)
So every real number is an image, and f is one-one and onto.