6. Let f(x)=x+4 and g(x)=(x-2)^{2}. Find a function u so that f(g(u(x)))=4 x^{2}-8 x+8.

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Step:1 ：First, we find the composition of functions f(g(x)).Step:2 ：f(g(x)) = f((x-2)^2) = (x-2)^2 + 4Step:3 ：Now, we want to find a function u(x) such that f(g(u(x))) = 4x^2 - 8x + 8.Step:4 ：We can rewrite the given function as:Step:5 ：4x^2 - 8x + 8 = (2x-4)^2Step:6 ：Comparing this with the expression for f(g(x)), we can see that u(x) = 2x-4.Step:7 ：So, the function u(x) is:Step:8 ：boxed{u(x) = 2x-4}

Answer

Step: 1 ：First, we find the composition of functions f(g(x)).Step: 2 ：f(g(x)) = f((x-2)^2) = (x-2)^2 + 4Step: 3 ：Now, we want to find a function u(x) such that f(g(u(x))) = 4x^2 - 8x + 8.Step: 4 ：We can rewrite the given function as:Step: 5 ：4x^2 - 8x + 8 = (2x-4)^2Step: 6 ：Comparing this with the expression for f(g(x)), we can see that u(x) = 2x-4.Step: 7 ：So, the function u(x) is:Step: 8 ：boxed{u(x) = 2x-4}

6. Let $f(x)=x+4$ and $g(x)=(x-2)^{2}$. Find a function $u$ so that $f(g(u(x)))=4 x^{2}-8 x+8$.

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