Question:
Mark (√) against the correct answer in the following:
If $f(x)=x^{2}-3 x+2$ then (f of) $(x)=?$
A. $x^{4}$
B. $x^{4}-6 x^{3}$
C. $x^{4}-6 x^{3}+10 x^{2}$
D. None of these
Solution:
$f(x)=x^{2}-3 x+2$
$\Rightarrow f(x)=x^{2}-2 x-x+2=x(x-2)-1(x-2)$
$\Rightarrow f(x)=(x-2)(x-1)$
$\Rightarrow f(x)=(x-2)(x-1)$
$\Rightarrow f(f(x))=(f(x)-2)(f(x)-1)$
$\Rightarrow f(f(x))=((x-2)(x-1)-2)((x-2)(x-1)-1)$
$\Rightarrow f(f(x))=\left(x^{2}-3 x+2-2\right)\left(x^{2}-3 x+2-1\right)$
$\Rightarrow f(f(x))=\left(x^{2}-3 x\right)\left(x^{2}-3 x+1\right)$
$\Rightarrow f(f(x))=x^{4}-3 x^{3}+x^{2}-3 x^{3}+9 x^{2}-3 x$
$\Rightarrow f(f(x))=x^{4}-6 x^{3}+10 x^{2}-3 x$