A composite function
It is given that f ( x ) = x 3 − 3 x + 1 . Find the number of real roots of f ( f ( x ) ) .
The answer is 7.
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2 solutions
f ( x ) = x 3 − 3 x + 1 f ′ ( x ) = 3 x 2 − 3 f ′ ( x ) = 0 at x = 1 , − 1 f ( − 1 ) = 3 , f ( 1 ) = − 1 For our final estimation, we see f ( − 2 ) < 0 , f ( 2 ) > 0 .
Thus the 3 roots of
f
(
x
)
=
0
are such that
−
2
<
α
<
−
1
0
<
β
<
1
1
<
γ
<
2
Now estimate the graph of
f
(
x
)
.
We could easily solve the problem without any rigorous calculation or actually solving the 9 degree resulting polynomial. Cheers!
suppose f(x) has A , B ,C as its roots, then f(f(x))=0 when f(x) = A,B or C. So , draw the graph of f(x) and then draw lines of Y=A, Y=B , Y=C. These will intersect at 7 points!!!