Let $f: \mathrm{R} \rightarrow \mathrm{R}$ be defined as $f(\mathrm{x})=\frac{2 \mathrm{x}^{2}-3 \mathrm{x}+2}{3 \mathrm{x}^{2}+\mathrm{x}+3}$. Then $f$ is :

JEE Main · 2026 · 06 Apr 2026 · Shift 2 · Sets Relations and Functions

Let

f : R \to R

be defined as

f (x) = \frac{2 x ^{2} - 3 x + 2}{3 x ^{2} + x + 3}

. Then

f

is :

Answer: (D)

Writing

y = \frac{2 x ^{2} - 3 x + 2}{3 x ^{2} + x + 3}

(the denominator stays positive for all

x

) and clearing,

(3 y - 2) x^{2} + (y + 3) x + (3 y - 2) = 0

. For a real

x

the discriminant must be non-negative:

(y + 3)^{2} - 4 (3 y - 2)^{2} \geq 0

, which factors as

(7 y - 1) (5 y - 7) \leq 0

, so

y \in [\frac{1}{7}, \frac{7}{5}]

. The range is a bounded proper subset of

R

, so

f

is not onto, and since values are attained at two distinct

x

the function is many-one. Hence

f

is neither one-one nor onto.

JEE Main · 2026 · 06 Apr 2026 · Shift 2 · Single correct option