First slide

Relations XII

Question

$Let f : R \to [0, \frac{π}{2}) defined by f (x) = \tan^{- 1} (x^{2} + x + a), then the set of values of a for which/is onto is$

Moderate

A

$[0, \infty)$
B

$[2, 1]$
C

$[\frac{1}{4}, \infty)$
D

None of these

Solution

$Since co-domain = [0, \frac{π}{2}) ∴ for f to be onto, range = [0, \frac{π}{2})$
$This is possible only when x^{2} + x + a \geq 0 T h e q u a d r a t i c e x p r e s s i o n i s p o s i t i v e f o r a l l r e a l v a l u e s, i f t h e d i s c r i m i n a n t i s n e g a t i v e$
$∴ 1^{2} - 4 a \leq 0 \Rightarrow a \geq \frac{1}{4}$

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