If the domain of the function f(x) = ⁻¹ ( x^2 + ax - 2 2x^2 + x + 3 ) is the set of all real numbers, then the number of integral values that a can take is

Step-by-step solution

The given function is f(x) = ⁻¹ ( x^2 + ax - 2 2x^2 + x + 3 ) . For the domain to be all real numbers, the argument of the inverse sine function must satisfy -1 x^2 + ax - 2 2x^2 + x + 3 1 for all x R . The denominator 2x^2 + x + 3 has a discriminant D = 1 - 24 = -23 0 for all real x . We can multiply the inequalities by the denominator without changing the signs. Case 1: Left inequality -(2x^2 + x + 3) x^2 + ax - 2 3x^2 + (a + 1)x + 1 0 For this to hold for all x R , its discriminant must be less than or equal to