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Question
Let
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
Find fof.
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Solution
f (x) =`{ (1 + x, 0≤ x ≤ 2) , (3 -x , 2 < x ≤ 3):}`
It can be written as,
f (x) = `{ (1 +x , 0 ≤ x ≤ 1) , (1 +x, 1< x ≤ 2) ,( 3 - x, 2 < x ≤ 3):}`
When, 0 ≤ x ≤ 1
Then , `f (x) = 1 +x `
Now when , 0 ≤ x ≤ 1 then ,1 ≤ x + 1 ≤ 2
Then , `f (f(x))` = 1 + (1 + x ) = 2 + x [ ∵ 1 ≤ f (x) < 2]
When ,1 < x ≤ 2
Then , f (x) = 1 + x
Now when , 1 < x ≤ 2 then,2 < x +1 ≤ 3
Then , f (f(x)) = 3 − ( 1+ x ) = 2 − x [ ∵ 2 ≤ f(x) <3 ]
When , 2 < x ≤ 3
Then , f (x) = 3 - x
Now when ,2< x ≤ 3 then ,0 ≤ 3 − x < 1
Then , f (f(x)) = 1 + ( 3 − x ) = 4 − x [ ∵ 0 ≤ f (x) < 1 ]
f(f(x)) = ` {(2 + x , 0 ≤ x ≤ 1) , (2 -x, 1 < x ≤ 2),( 4- x , 2 < x ≤ 3):}`
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