If f is a function defined as f ( x ) = x 2 − x + 5 , f : ( 1 2 , ∞ ) → ( 19 4 , ∞ ) , and g ( x ) is its inverse function, then g ' 7 is equal to

Step-by-step solution

f ⁡ x = x 2 - x + 5 f ' ( x ) = 2 x - 1 ⇒ for y = 7 , x > 1 2 ∴ 7 = x 2 - x + 5 ⇒ x 2 - x - 2 = 0 ⇒ x - 2 x + 1 = 0 ⇒ x = - 1, 2 ⇒ x = 2 (as x = - 1 is rejected) I f f x & g x a r e i n v e r s e o f e a c h o t h e r, g f x = x ⇒ g ' f x . f ' x = 1 substituting x = 2 in this equation, we get, g ' ( f ( 2 ) ) . f ' ( 2 ) = 1 ⇒ g ' ( f ( 2 ) ) = 1 f ' ( 2 ) g ' ( 7 ) = 1 2 ( 2 ) - 1 = 1 3 ( at x = 2 )