Let f , g : ( 1 , ∞ ) → R be defined as f ( x ) = 5 x + 2 2 x + 3 and g ( x ) = 1 − x 2 − 3 x . If the range of the function f ∘ g : [ 2 , 4 ] → R is [ α , β ] , then β − α 1 is equal to

Question

Let f , g : ( 1 , ∞ ) → R be defined as f ( x ) = 5 x + 2 2 x + 3 ​ and g ( x ) = 1 − x 2 − 3 x ​ . If the range of the function f ∘ g : [ 2 , 4 ] → R is [ α , β ] , then β − α 1 ​ is equal to

TestHub · Accepted Answer

​ fog ( x ) = f ( g ( x )) = f ( 1 − x 2 − 3 x ​ ) = 5 ( 1 − x 2 − 3 x ​ ) + 2 2 ( 1 − x 2 − 3 x ​ ) + 3 ​ = 10 − 15 x + 2 − 2 x 4 − 6 x + 3 − 3 x ​ = ( 12 − 17 x 7 − 9 x ​ ) ∴ [ 12 − 7 x  = 0 x  = 17 12 ​ ​ [ fog ( 2 ) = 12 − 17 ( 2 ) 7 − 9 ( 2 ) ​ = − 22 − 11 ​ = 2 1 ​ fog ( 4 ) = 12 − 17 ( 4 ) 7 − 9 ( 4 ) ​ = − 56 − 29 ​ = 56 29 ​ ​ Range of fog : [ α , β ] = [ 2 1 ​ , 56 29 ​ ] ∴ ( β − α ) = 56 29 ​ − 2 1 ​ = 56 29 − 28 ​ = 56 1 ​ ( β − α ) 1 ​ = 56 ​

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