Question

If `f(x)=-1+|x-2|,0lexle4`
`g(x)=2-|x|,-1lexle3`
Then find `fog(x)` and `gof(x)`. Also draw their rough sketch.

Answer 1

`fog(x)={-1+|g(x)-2|,0leg(x)le4,-1lexle3`
`=+-1+|-2|+x|-2|,0le2-|x|le4,-1lexle3`
`=|-1+|x|,-2lexle2,-1lexle3`

`={(-(1+x),-1lexle0),(x-1,0ltxle2):}`

`gof(x)={2-f(x)|f(x)|,-1lef(x)3,0lexle4`
`={2-|-1_|x-2||,-1le-1+|x-2|le3,0 lexle4`
`={2-|-1+|x-2||,-2lexle6,0lexle4`
`={(x+1,0lexlt1),(3-x,1lexle2),(x-1,2ltxle3),(5-x,3ltxle4):}`