Sets, Relations & Functions
One-One / Many-One Functions
nta_pyq_2023_jan
Grade None
Question:
Let $f: \mathbb{R} \to \mathbb{R}$ be a function such that $f(x) = \dfrac{x^2 + 2x + 1}{x^2 + 1}$. Then
$f(x)$ is many-one in $(-\infty, -1)$
$f(x)$ is many-one in $(1, \infty)$
$f(x)$ is one-one in $[1, \infty)$ but not in $(-\infty, \infty)$
$f(x)$ is one-one in $(-\infty, \infty)$
Step-by-Step Solution
Key Concept: $f(x)=1+\frac{2}{x+1/x}$; analyze monotonicity using derivative and identify turning point at $x=-1$.
$f'(x)=\frac{2(1-x^2)}{(x^2+1)^2}$. $f$ increases on $(-1,1)$ and decreases outside, so it is many-one on $\mathbb{R}$ but one-one on $[1,\infty)$.
Correct Answer: 3