Sets, Relations & Functions
One-One and Many-One — Statement Based
nta_pyq_2026_jan
Grade None
Question:
Given below are two statements:
**Statement I:** The function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\dfrac{x}{1+|x|}$ is one-one.
**Statement II:** The function $f:\mathbb{R}\to\mathbb{R}$ defined by $f(x)=\dfrac{x^2+4x-30}{x^2-8x+18}$ is many-one.
In the light of the above statements, choose the correct answer:
Both Statement I and Statement II are true
Statement I is true but Statement II is false
Statement I is false but Statement II is true
Both Statement I and Statement II are false
Step-by-Step Solution
Key Concept: Statement I: For $x\geq0$, $f'(x)=\tfrac{1}{(1+x)^2}>0$; for $x<0$, $f'(x)=\tfrac{1}{(1-x)^2}>0$. Continuous at $x=0$, so strictly increasing $\Rightarrow$ one-one. Statement II: $f'(x)=0\Rightarrow x=4\pm\sqrt{2}$, $f'$ changes sign $\Rightarrow$ not monotonic $\Rightarrow$ many-one.
Both Statement I and Statement II are true.
Correct Answer: 1