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# Help with functions!!

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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.

MIRB16  Jul 21, 2017
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#1
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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.

geno3141  Jul 21, 2017

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