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if f:R->[pi/6,pi/2),f(x)= arcsin (x^2 - a/x^2 + 1) is a onto function,den set of values of "a" is.....

if f:R->[pi/6,pi/2),f(x)= arcsin (x^2 - a/x^2 + 1) is a onto function,den set of values of "a" is.....

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1 Answers

Jitender Singh IIT Delhi
askIITians Faculty 158 Points
9 years ago
Ans:
y = sin^{-1}(x^{2}-\frac{a}{x^{2}}+1)
(x^{2}-\frac{a}{x^{2}}+1)=siny
x^{4}+(1-siny)x^{2}-a=0
This is simple Quadratic in x2,
\Delta \geq 0
(1-siny)^{2}\geq 4.(-a)
sin^{2}y-2siny+1+4a\geq 0
\Delta <0
4<4(4a+1)
a>0
Thanks & Regards
Jitender Singh
IIT Delhi
askIITians Faculty

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