(i) If f(x) = {(1-x, 0≤x≤1)(1+x, 1<x≤2) then which of the following is not true (a) f is continuous in (0,1) (b) f is continuous in (1, 2 )

Question

(i) If f(x) = \(\begin{cases} 1-x,& \quad 0≤x≤1\\ 1+x,& \quad 1<x≤2 \end{cases} \) then which of the following is not true

(a) f is continuous in ( 0, 1 )
(b) f is continuous in (1, 2 )
(c) f is continuous in [ 0, 2 ]
(d) f is continuous in [ 0,1 ]

(ii) Find f(3⁺) and f(5^–)
(iii) Hence find the value of ‘a’ and ‘b’ so that f(x) is continuous.

Answer 1

(i) (c) Since f is not continuous at x = 1.

(ii) 

(iii) Since f (x) is continuous, it is continuous at x = 3 and x = 5
∴ f(3⁺) = f(3) ⇒ 3a + b = 1 ____(1)
and f(5^–) = f(5) ⇒ 5a + b = 7 ____(2)
(2) – (1) ⇒ 2a = 6, a = 3
(1) ⇒ b = 1 – 3 a ⇒ b = -8
∴ a = 3, b = – 8.