JEE Main 2026 Apr 6 Shift 2, Mathematics Q1: Types of Functions

JEE Main2026Apr 6, Shift 2Mathematics

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:

A
both one-one and onto
B
one-one but not onto
C
onto but not one-one
D
neither one-one nor onto

Solution

Set $y = \frac{2 x ^{2} - 3 x + 2}{3 x ^{2} + x + 3}$ and rearrange:

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

For real $x$ , the discriminant must be non-negative:

$(y + 3)^{2} - 4 (3 y - 2)^{2} \geq 0$ $\Rightarrow$ $(7 y - 1) (5 y - 7) \leq 0$ $\Rightarrow$ $y \in [\frac{1}{7}, \frac{7}{5}]$ .

So the range is a bounded subset of $R$ , hence $f$ is not onto.

Since $f$ achieves both its maximum and minimum on $R$ but is continuous and non-monotonic, every interior range value is attained at more than one $x$ . Hence $f$ is not one-one.

$∴$ $f$ is neither one-one nor onto.

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JEE Main 2026 Apr 6 Shift 2, Mathematics Q1: Types of Functions

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