Let f ( x ) = lo g e x and g ( x ) = 2 x 2 − 2 x + 1 x 4 − 2 x 3 + 3 x 2 − 2 x + 2 . Then the domain of f ∘ g is

Question

Let f ( x ) = lo g e ​ x and g ( x ) = 2 x 2 − 2 x + 1 x 4 − 2 x 3 + 3 x 2 − 2 x + 2 ​ . Then the domain of f ∘ g is

TestHub · Accepted Answer

f ( g ( x )) = ln ( 2 x 2 − 2 x + 1 x 4 − 2 x 3 + 3 x 2 − 2 x + 2 ​ ) Since 2 x 2 − 2 x + 1 > 0 ∀ x ∈ R ∵ ( − 2 ) 2 − 4 ( 2 ) < 0 Consider ​ x 4 − 2 x 3 + 3 x 2 − 2 x + 2 = ( x 4 − 2 x 3 + x 2 ) + ( x 2 − 2 x + 1 ) + ( 1 + x 2 ) = x 2 ( x − 1 ) 2 + ( x − 1 ) 2 + ( x 2 + 1 ) > 0∀ x ∈ R ⇒ g ( x ) > 0∀ x ∈ R ⇒ ln f (( x )) , f ( x ) > 0∀ x ∈ R ⇒ x ∈ R is domain ​

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