f(x)= -1, -2 ≤ x ≤ 0 x -1 , 0 < x ≤ 2 g(x) = f(│x│) + │f(x)│ find g(x)

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Blazing goIITian

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f(x) = -1, -2 ≤ x ≤ 0

= x -1, 0 < x ≤ 2

f(x) < 0 for -2 <=x<1, and f(x) >=0 for 1<=x<=2

Concept: |x| = x, if x >= 0

= -x if x < 0

Now..

g(x) = f( |x| ) + |f(x)|

Case 1: For x>=0, f(x)>=0. ( for 1<=x<=2 )

g(x) = f(x) + f(x) = 2(fx) = 2(x-1) = 2x - 2

Case 2: For x>=0, f(x) <0 ( for 0<=x<1 )

g(x) =f(x) - f(x) = 0

Case 3: For x<0, f(x)>=0 ( for NO value of x )

g(x) = f(-x) + f(x) CASE REJECTED

Case 4: For x<0, f(x)< 0 ( for -2<=x<=0 )

g(x) = f(-x) - f(x) = (-x-1) - (-1) = -x

Hence,

g(x) = -x for -2<=x<0.

g(x) = 0 for 0<=x<1

g(x) = 2x-2 for 1<=x<=2

Everyone's Stupid.

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