Set, Relation and FunctionMedium
Question
Let $f:R \rightarrow R$ be defined as $f(x) = \frac{2x^{2} - 3x + 2}{3x^{2} + x + 3}$. Then $f$ is :
Options
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
Sol. [Key Concept: Set $\frac{2x^{2} - 3x + 2}{3x^{2} + x + 3} = y$ and rearrange to a quadratic in $x$; a real $x$ needs discriminant $\geq 0$, which bounds $y$ (so $f$ is into) and shows $f$ is non-monotonic (many-one).]
$${\ \frac{2x^{2} - 3x + 2}{3x^{2} + x + 3} = y }{(3y - 2)x^{2} + (y + 3)x + (3y - 2) = 0 }{D \geq 0 }{(7y - 1)(5y - 7) \leq 0 }{y \in \left\lbrack \frac{1}{7},\frac{7}{5} \right\rbrack }$$Hence $f(x)$ is into and $f(x)$ is many-one because $f(x)$ is non-monotonic
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