Given below are two statements: Statement I: The function f: R...

Statement I: $f(x) = \frac{x}{1+|x|}$. For $x \geq 0$, $f(x) = \frac{x}{1+x}$ is strictly increasing. For $x < 0$, $f(x) = \frac{x}{1-x}$ with $f'(x) = \frac{1}{(1-x)^2} > 0$, also strictly increasing. The function is continuous at $x = 0$, so $f$ is strictly increasing on $\mathbb{R}$ and hence one-one. Statement I is true.
Statement II: $f(x) = \frac{x^2+4x-30}{x^2-8x+18}$. The denominator $(x-4)^2 + 2 > 0$ for all $x$.
$f'(x) = \frac{-12(x^2 - 8x + 14)}{(x^2 - 8x + 18)^2}$
$f'(x) = 0 \Rightarrow x = 4 \pm \sqrt{2}$. Since $f'$ changes sign, $f$ is not monotonic and hence many-one. Statement II is true.
Both statements are true.