Given a function f(x)= array ccc -1 & If & x 0 a x+b & If & 0 x 1 1 & If & x 1 array . where a , b are constants. The function is continuous everywhere. What is the value of b?

Step-by-step solution

aligned & F ( x )= array ccc -1 & , & x 0 ax + b & , & 0 x 1 1, & x 1 array . & At x=0 , & L.H.L. = R.HL. & _ x 0⁻ f(x)= _ x 0⁺ f(x) & -1= _ x 0 (a x+b) & -1=b aligned Since the given function is also continuous at x = 1 L.H.L. = R.H.L. aligned & _ x 1⁻ f(x)= _ x 1⁺ f(x) & a+b=1 & a-1=1 & a=1+1=2 aligned Function is continuous at x = 0 aligned & _ x 0⁻ f(x)= _ x 0⁺ f(x) & -1=a(0)+b b=-1 aligned Now, f ( x ) is also continuous at x =1 aligned & _ x 1⁻ f(x)= _ x 1⁺ f(x) & a(1)+b=1 & a=1-b=1+1 & a=2 . aligned b = –1.