# Composite functions

• Jul 25th 2007, 08:57 PM
Composite functions
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
• Jul 25th 2007, 09:50 PM
earboth
Quote:

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\$
• Jul 25th 2007, 09:54 PM
tukeywilliams
Whoops, didnt see that \$\displaystyle g(x) \$ was given.
• Jul 25th 2007, 10:06 PM
red_dog
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\$