Suppose that $g(x)=4 x+4$ and $h(x)=16 x^{2}+32 x+25$ Find a function $f$, such that $f(g(x))=h(x)$. (Think about what operations you would have to perform on the formula for $g$ to end up with the formula for $h$

Step 1 ：Given that \(g(x)=4x+4\) and \(h(x)=16x^2+32x+25\), we can rewrite \(h(x)\) as follows: \(h(x)=16x^2+32x+25 = 4(4x^2+8x+6.25)\).

Step 2 ：Now, we can see that \(4x^2+8x+6.25\) is a perfect square trinomial, which can be written as \((2x+2.5)^2\). So, \(h(x)=4(2x+2.5)^2\).

Step 3 ：Now, we can see that \(2x+2.5\) is equal to \(0.5g(x)\), so we can substitute \(0.5g(x)\) into the equation for \(h(x)\): \(h(x)=4(0.5g(x))^2\).

Step 4 ：Therefore, the function \(f\) such that \(f(g(x))=h(x)\) is \(f(x)=4(0.5x)^2\).

Step 5 ：To check our answer, we can substitute \(g(x)\) into \(f(x)\) and see if we get \(h(x)\): \(f(g(x))=4(0.5(4x+4))^2=4(2x+2)^2=16x^2+32x+16=16x^2+32x+25\).

Step 6 ：So, our answer is correct. The function \(f\) such that \(f(g(x))=h(x)\) is \(\boxed{f(x)=4(0.5x)^2}\).