Composite functions

**madman1611** · July 25th 2007, 08:57 PM

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

**earboth** · July 25th 2007, 09:50 PM

Originally Posted by madman1611

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 $f(g(x)) = (x-2)^2+4$

2. $f(g(u(x))) = 4x^2-8x+8 = 4x^2-8x+{\color{red}+4+4} = (2x-2)^2+4$

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 $u(x) = 2x$

**tukeywilliams** · July 25th 2007, 09:54 PM

Whoops, didnt see that $g(x)$ was given.

**red_dog** · July 25th 2007, 10:06 PM

There is another solution.
$f(g(x))=x^2-4x+8\Rightarrow f(g(u(x)))=u^2(x)-4u(x)+8$
So we have to solve the functional equation $u^2(x)-4u(x)+8=4x^2-8x+8$
which is equivalent to $[u(x)-2x][u(x)+2x-4]=0$ .
Then $u(x)=2x$ or $u(x)=-2x+4$

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