Let f:R [0,pi/2] defined by f(x)=tan-1(x^2+x+a), then find the value or set of values of 'a' for which f - Maths - Relations and Functions

Question

Let f:R? [0,pi/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

Aarushi Mishra · Accepted Answer

f:R->0,π2f(x)=tan-1(x2+x+a)We know tan-1y will lie in 0,π2 if y≥0for f(x) to be onto when x2+x+a≥0 and x2+x+a will take all values upto infinityNote as x->∞ then (x2+x+a) ->∞ and it is continuous therefore it will take all values upto infinity
 Also x2+x+a should take value 0 for atleast one real xTherefore we only need to find the value of a such that x2+x+a=0 and x2+x+a≥0______________________For quadritic equationax2+bx+cif a>0 and D=b2 - 4ac=0 thenx2+x+a≥0and will take value 0 too
______________________a>0D=12-4a=04a=1a=14

Let f:R? [0,pi/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.