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

$\left[\frac{1}{4},∞\right)$

Since codomain $\left[0,\frac{π}{2}\right)$

∴ for f to be onto, range = $\left[0,\frac{π}{2}\right)$

This is possible only when $x^2+x+a≥0$

$∴ 1^2-4a≤0⇒a≥\frac{1}{4}$

∴ The set of values of a for which f is onto is $\left[\frac{1}{4},∞\right)$