Need help with this one (composite functions):
Let f(x) = x + 4 and g(x)= (x - 2)^2 . Find a function u so that f(g(u(x))) = 4x^2 - 8x + 8
Hello,
1. Calculate $\displaystyle f(g(x)) = (x-2)^2+4$
2. $\displaystyle f(g(u(x))) = 4x^2-8x+8 = 4x^2-8x+{\color{red}+4+4} = (2x-2)^2+4$
3. Now compare the equation from 1. with the result from 2. Obviously the term 2x was plugged into the equation #1 instead of x. Thus $\displaystyle u(x) = 2x$
There is another solution.
$\displaystyle f(g(x))=x^2-4x+8\Rightarrow f(g(u(x)))=u^2(x)-4u(x)+8$
So we have to solve the functional equation $\displaystyle u^2(x)-4u(x)+8=4x^2-8x+8$
which is equivalent to $\displaystyle [u(x)-2x][u(x)+2x-4]=0$.
Then $\displaystyle u(x)=2x$ or $\displaystyle u(x)=-2x+4$