How to work out FoF, the composite of this function: f(n)={1/3+2x, 0<=x<=1/3 2-3x, 1/3<=x<=2/3 2x-4/3, 2/3<=x<=1

pobi1k

pobi1k

Answered question

2022-09-10

How to work out F o F, the composite of this function:
f ( n ) = { 1 3 + 2 x , 0 x 1 3 2 3 x , 1 / 3 x 2 3 2 x 4 3 , 2 3 x 1

Answer & Explanation

trabadero2l

trabadero2l

Beginner2022-09-11Added 15 answers

Think about what happens to each value of x as it passes through f f, and which questions you have to ask about it. For example, we first have to ask "is x less than 1 / 3, between 1 / 3 and 2 / 3, or greater than 2 / 3?"
If 0 x 1 / 3, then we use the first line and f ( x ) = 1 3 + 2 x. Remember, x is between 0 and 1 / 3, so 1 3 + 2 x is between 1 / 3 and 1 - which means that on the next pass, we use either the second or third lines. We use the second line if f ( x ) is between 1 / 3 and 2 / 3, and the third line otherwise. So the first two pieces of our final composited function are:
2 3 ( 1 3 + 2 x ) if 0 x 1 / 3 and 1 / 3 1 3 + 2 x 2 / 3
2 ( 1 3 + 2 x ) 4 3 if 0 x 1 / 3 and 2 / 3 1 3 + 2 x 1

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