Let f(x) = x^2 - 2x . Find all real numbers x such that f(x) = f(f(x))?
List your solutions separated by commas from least to greatest.
If f(x) = x2 - 2x, find all real numbers x such that f(x) = f( f(x) ).
f( f(x) ) = f( x2 - 2x ) = ( x2 - 2x )2 - 2( x2 - 2x )
Since f(x) = f( f(x) ) ---> x2 - 2x = ( x2 - 2x )2 - 2( x2 - 2x )
---> x2 - 2x + 2( x2 - 2x ) = ( x2 - 2x )2
---> 3( x2 - 2x ) = ( x2 - 2x )2
Case 1: If x2 - 2x <> 0 ---> 3( x2 - 2x ) = ( x2 - 2x )2
divide both sides by x2 - 2x ---> 3 = x2 - 2x
---> x2 - 2x - 3 = 0
---> (x - 3)(x + 1) = 0
---> x = 3 or x = -1
Case 2: If x2 - 2x = 0 ---> x(x - 2) 0
---> x = 0 or x = 2
These are the four possible answers.