Let f : R → R and g : R → R be defined as f x = x + a , x < 0 | x - 1 | , x ≥ 0 and g x = x + 1 , x < 0 ( x - 1 ) 2 + b , x ≥ 0 , where a , b are non-negative real numbers. If g o f ( x ) is continuous for all x ∈ R , then a + b is equal to ______ .

Step-by-step solution

g f x = f ( x ) + 1 f ( x ) < 0 ( f ( x ) - 1 ) 2 + b f ( x ) ≥ 0 g f x = x + a + 1 x + a < 0 ;   x < 0 | x - 1 | + 1 | x - 1 | < 0 ;   x ≥ 0 ( x + a - 1 ) 2 + b x + a ≥ 0 ;   x < 0 ( | x - 1 | - 1 ) 2 + b | x - 1 | ≥ 0 ;   x ≥ 0 g f x = x + a + 1 x ∈ ( - ∞ , - a ) ;   x ∈ ( - ∞ , 0 ) | x - 1 | + 1 x ∈ ϕ ( x + a - 1 ) 2 + b x ∈ [ - a , ∞ ) ;   x ∈ ( - ∞ , 0 ) ( | x - 1 | - 1 ) 2