Question 14 (CSIR NET June 2019)

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous and one-one function, then-

(a) $f$ is onto

(b) $f$ is either strictly increasing or strictly decreasing

(c) There exist $ x \in \mathbb{R}$ s.t. $f(x)=1$

(d) $f$ is unbounded.

Explanation: (b) is correct.

Consider $f(x) = e^x$ then $f$ is not onto $\implies$ (a) is false.

Consider $f(x) = 1+e^x$ then (c) is false.

Consider $f(x) = \frac{1}{1+e^x}$, as $0 < e^x < \infty \implies 1 < 1+e^x < \infty$

$\implies 0 < \frac{1}{1+e^x} < 1 \implies f(x)$ is bounded $\implies$ (d) is false.

$\implies$ Option b) is true.