If $f : R \rightarrow R$ is defined by $f(x) = x^{2} - 3x + 2$,find $f(f(x))$.
Given $f(x) = x^{2} - 3x + 2$.
To find $f(f(x))$,we substitute $f(x)$ into the function $f$:
$f(f(x)) = f(x^{2} - 3x + 2)$
$= (x^{2} - 3x + 2)^{2} - 3(x^{2} - 3x + 2) + 2$
$= (x^{4} + 9x^{2} + 4 - 6x^{3} + 4x^{2} - 12x) - 3x^{2} + 9x - 6 + 2$
$= x^{4} - 6x^{3} + (9x^{2} + 4x^{2} - 3x^{2}) + (-12x + 9x) + (4 - 6 + 2)$
$= x^{4} - 6x^{3} + 10x^{2} - 3x$
To find $f(f(x))$,we substitute $f(x)$ into the function $f$:
$f(f(x)) = f(x^{2} - 3x + 2)$
$= (x^{2} - 3x + 2)^{2} - 3(x^{2} - 3x + 2) + 2$
$= (x^{4} + 9x^{2} + 4 - 6x^{3} + 4x^{2} - 12x) - 3x^{2} + 9x - 6 + 2$
$= x^{4} - 6x^{3} + (9x^{2} + 4x^{2} - 3x^{2}) + (-12x + 9x) + (4 - 6 + 2)$
$= x^{4} - 6x^{3} + 10x^{2} - 3x$
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