Sketch all the continuous functions f:R->R which satisfy (f(x))^2=(x−1)^2(x−2)^2. Justify your answers.

Marcelo Maxwell

Marcelo Maxwell

Answered question

2022-09-20

Question: Sketch all the continuous functions f : R R which satisfy
( f ( x ) ) 2 = ( x 1 ) 2 ( x 2 ) 2 .
Justify your answers.
I have found the eight possible continuous functions as follows:
f ( x ) = ( x 1 ) ( x 2 ) f ( x ) = ( x 1 ) ( x 2 ) f ( x ) = | x 1 | ( x 2 ) f ( x ) = | x 1 | ( x 2 ) f ( x ) = ( x 1 ) | x 2 | f ( x ) = ( x 1 ) | x 2 | f ( x ) = | x 1 | | x 2 | f ( x ) = | x 1 | | x 2 | .
and have got the conclusion of this question which has at most eight possible such functions, but how to proof the "at most" statement.

Answer & Explanation

Abagail Stephenson

Abagail Stephenson

Beginner2022-09-21Added 9 answers

Any such function f has to vanish at x = 1 and x = 2 and nowhere else. Therefore the sign of f has to be constant in each of the intervals   ] , 1 [   ,   ] 1 , 2 [   ,   ] 2 , [  . It follows that you have exactly two choices for the graph of f in each of these intervals. As the value at the junctions is 0 it follows that f will automatically be continuous for any choice made. In particular you are allowed to make the choices independently. This gives a total of 2 3 = 8 possibilities.

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