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 ______ .

Question

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 ______ .,

MARKS App · Accepted Answer

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 + b x ∈ R ; x ∈ [ 0 , ∞ ) g f x = x + a + 1 x ∈ ( - ∞ , - a ) ( x + a - 1 ) 2 + b x ∈ [ - a , 0 ) ( | x - 1 | - 1 ) 2 + b x ∈ [ 0 , ∞ ) g ( f x ) is continuous at x = - a & at x = 0 1 = b + 1 & ( a - 1 ) 2 + b = b b = 0 & a = 1 ⇒ a + b = 1

Search any question & find its solution

Looking for more such questions to practice?