Question

If $f:R\to R$ is defined by f(x) = $x^2-3x+2$, find $f\left(f\right(x\left)\right)$.

Answer 1

Solve for $f\left(f\right(x\left)\right)$.
Given: $f\left(x\right)=x^2-3x+2$

$f\left(f\right(x\left)\right)=f{\left(x\right)}^2-3f\left(x\right)+2$

$={(x^2-3x+2)}^2-3(x^2-3x+2)+2$

Using formula ${(a-b+c)}^2=a^2+b^2+c^2-2ab+2ac-2bc$

$={\left(x^2\right)}^2+{(-3x)}^2+2^2-2x^2\left(3x\right)+2x^2\left(2\right)$

$-2\left(3x\right)\left(2\right)-3(x^2-3x+2)+2$

$=x^4+9x^2+4-6x^3-12x+4x^2-3x^2+9x-6+$

$=x^4-6x^3+9x^2+4x^2-3x^2-12x+9x-6+2+4$

$=x^4-6x^3+10x^2-3x$