Let f(x) = _e x and g(x) = x^4 - 2x^3 + ax^2 - 2x + 1 x^2 + 2x + 5 . If the domain of the function f g is R , then the set of all possible values of a is

Step-by-step solution

The domain of f(g(x)) = _e(g(x)) is the set of all x for which g(x) > 0 . Given that the domain is R , we must have: g(x) = x^4 - 2x^3 + ax^2 - 2x + 1 x^2 + 2x + 5 > 0 x R For the denominator, the discriminant is D = 2^2 - 4(1)(5) = -16 0 for all x R . This implies that the numerator must be strictly positive for all x R : x^4 - 2x^3 + ax^2 - 2x + 1 > 0 For x = 0 , the inequality gives 1 > 0 , which is true. For x 0 , dividing by x^2 gives: x^2 - 2x + a - 2 x + 1 x^2 > 0 (x^2 + 1 x^2 ) - 2 (x + 1 x ) + a > 0 Let t