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Let f(x) be defined on [-2,2] and be giv...

Let `f(x)` be defined on `[-2,2]` and be given by
`f(x)={(-1",",-2 le x le 0),(x-1",",0 lt x le 2):} and g(x)=f(|x|) +|f(x)|`.
Then find `g(x)`.

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We have
`f(x)={(-1",",-2 le x le 0),(x-1",",0 lt x le 2):}`
or `f(|x|)={(-1",",-2 le |x| le 0),(|x|-1",",0 le |x| le 2):}`
`=|x|-1,0 le |x| le 2`
` " " `(As `-2 le |x| lt 0` is not possible)
`={(-x-1",",-2 le x le 0),(x-1",",0 lt x le 2):} " (1)" `
Again, `f(x)={(-1",",-2 le x le 0),(x-1",",0 lt x le 2):}`
or `|f(x)|={(|-1|",",-2 le x le 0),(|x-1|",",0 lt x le 2):}`
or `|f(x)|={(1",",-2 le x le 0),(-(x-1)",",0 lt x le 1),(+(x-1)",",1 lt x le 2):} " (2)" `
Therefore, `g(x)=f(|x|)+|f(x)|` can be expressed as
`g(x)={((-x-1)+1",",-2 le x le 0),((x-1)+(1-x)",",0 lt x le 1),((x-1)+(x+1)",",1 lt x le 2):}` [Using (1) and (2) ]
`={(-x",",-2 le x le 0),(0",",0 lt x le 1),(2(x-1)",",1 lt x le 2):}`
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