SOLUTION: Let f(x) = x^2 - 2x. Find all real numbers x such that f(x) = f(f(x)).

Question 1089842: Let f(x) = x^2 - 2x. Find all real numbers x such that f(x) = f(f(x)).
Answer by jim_thompson5910(35256) (Show Source): You can put this solution on YOUR website!
f(x) = x^2 - 2x
f(f(x)) = (f(x))^2 - 2*(f(x)) ... replace every x with f(x)
f(f(x)) = (x^2 - 2x)^2 - 2*(x^2 - 2x) ... replace the f(x) on the right side with x^2-2x
f(f(x)) = x^4 - 4x^3 + 4x^2 - 2*(x^2 - 2x)
f(f(x)) = x^4 - 4x^3 + 4x^2 - 2x^2 + 4x
f(f(x)) = x^4 - 4x^3 + 2x^2 + 4x
-------------------------------------
f(x) = f(f(x))
x^2 - 2x = x^4 - 4x^3 + 2x^2 + 4x
0 = x^4 - 4x^3 + x^2 + 6x
x^4 - 4x^3 + x^2 + 6x = 0
x(x^3 - 4x^2 + x + 6) = 0
x = 0 or x^3 - 4x^2 + x + 6 = 0
Solve x^3 - 4x^2 + x + 6 = 0 using a graphing calculator to get the following results: x = -1, x = 2, x = 3 (note how these values are roots, or x intercepts)
-------------------------------------
-------------------------------------
So overall, the four solutions are:
x = 0
x = -1
x = 2
x = 3

RELATED QUESTIONS
Let f(x) = x^2 - 2x. Find all real numbers x such that f(x) =... (answered by robertb,richard1234,Natolino1983)

Let f(x) = 3x^2 - 4x. Find the constant k such that f(x) = f(k - x) for all real numbers... (answered by richard1234)

Let f(x) = 3x^2 - 4x. Find the constant k such that f(x) = f(k - x) for all real numbers (answered by reviewermath)

Let f(x) = 3x^2 - 4x. Find the constant k such that f(x) = f(k - x) for all real numbers (answered by ikleyn)

Find the real number k such that f(x) = f(k - x) for all real numbers x, where f(x) =... (answered by ikleyn)

Let f be a function such that f(x) + f(2x + y) + 5xy = f(4x - y) - x^2 + 5xy - 8x + 17y... (answered by CPhill,ikleyn)

Find all real values of x such that f(x)=0 f(x)=49x^2-9 (answered by jim_thompson5910)

find all real values of x such that f(x)=0:... (answered by chibisan)

Let f be a function such that f(xy) + x = xf(y) + f(x) + xy^2 for all real numbers x... (answered by CPhill,ikleyn,Edwin McCravy,mccravyedwin)