Functions Part 2

Concept:A function is one‑one if it is strictly monotonic (always increasing or always decreasing).A function is many‑one if different inputs map to the same output.Explanation:For Statement I: $f(x)=\frac{x}{1+|x|}$.For $x \ge 0$: $f(x)=\frac{x}{1+x}$; derivative $f'(x)=\frac{1}{(1+x)^2}>0$.For $x < 0$: $f(x)=\frac{x}{1-x}$; derivative $f'(x)=\frac{1}{(1-x)^2}>0$.The derivative is positive for all $x \in \mathbb{R}$, so $f$ is strictly increasing.A strictly increasing function is one‑one. Hence Statement I is true.For Statement II: $f(x)=\frac{x^2+4x-30}{x^2-8x+18}$.Denominator discriminant: $D = (-8)^2-4\cdot1\cdot18 = 64-72 = -8 < 0$.So denominator never zero; function is continuous on $\mathbb{R}$.As $x \to \pm\infty$, $f(x) \to 1$ (ratio of leading coefficients).Because the function approaches 1 at both ends, it must have at least one local extremum.Thus a horizontal line (e.g. $y=1$) will intersect the graph at more than one point.Alternatively, set $y=\frac{x^2+4x-30}{x^2-8x+18}$ and cross‑multiply:$y(x^2-8x+18)=x^2+4x-30 \Rightarrow (y-1)x^2 + (-8y-4)x + (18y+30)=0$.For many values of $y$, this quadratic has two distinct real roots, proving the function is many‑one.Hence Statement II is true.Answer:Both Statement I and Statement II are true. (Option C)