If f ( x ) = x 2 + x + 4 3 and g ( x ) = x 2 + a x + 1 be two real functions, then the range of a for which g ( f ( x )) = 0 has no real solution is ____

Question

If f ( x ) = x 2 + x + 4 3 ​ and g ( x ) = x 2 + a x + 1 be two real functions, then the range of a for which g ( f ( x )) = 0 has no real solution is ____

TestHub · Accepted Answer

f ( x ) = x 2 + x + 4 3 ​ = ( x + 2 1 ​ ) 2 + 2 1 ​ ≥ 2 1 ​ g ( f ( x )) = ( f ( x ) ) 2 + a f ( x ) + 1 , for g ( f ( x )) = 0 a = − ( f ( x ) + f ( x ) 1 ​ ) ≤ − 2 ∴ If a > − 2 , g ( f ( x )) = 0 has no solutions

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