For some a , b , c ∈ ℕ , let f x = a x - 3 and g x = x b + c , x ∈ ℝ . If f o g - 1 x = x - 7 2 1 3 , then f ∘ g a c + g ∘ f b is equal to _____ .

Step-by-step solution

Given: f x = a x - 3 g x = x b + c , x ∈ ℝ f o g - 1   x = x - 7 2 1 3 Now, let h x = f ∘ g x ⇒ h - 1 x = x - 7 2 1 3 Let y = x - 7 2 1 3 ⇒ y 3 = x - 7 2 ⇒ x = 2 y 3 + 7 So, inverse of h - 1 x is 2 x 3 + 7 i.e., h x = f o g x = 2 x 3 + 7 Also, f o g x = a x b + c - 3 ⇒ a x b + a c - 3 = 2 x 3 + 7 On comparing, we get a = 2 ,   b = 3 , c = 5 So, f x = 2 x - 3 g x = x 3 + 5 Now, f o g   a c = f o g 10 = f 1005 = 2007 g o f b = g o f 3 = g 3 = 32 Therefore, f o