​ Let f:R →[0,π/2) defined by f(x) = tan^(–1) (x^2 + x + a), then find the value or set of values of ‘a’ for which f is onto.

Question

Let \(f:R\longrightarrow[0,\frac{\pi}{2})\) defined by f(x) = tan^–1 (x² + x + a), then find the value or set of values of ‘a’ for which f is onto.

Answer 1

Given function is 

\(f:R\longrightarrow[0,\frac{\pi}{2})\)

Since, f is onto ⇒ Range of f is \([0,\frac{\pi}{2})\)

It is possible only when a = \(\frac{1}{4}\)

As 

Hence, the required value of a = \(\frac{1}{4}\)