Find a formula for the inverse of the following function on the given domain, if possible.
\[
g(x)=(x-4)^{4}+4, x \geq 4
\]
Answer
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\[
g^{-1}(x)=
\]
does not have an inverse function

Step 1 ：The function given is \(g(x)=(x-4)^{4}+4\), with the domain \(x \geq 4\).

Step 2 ：This is a polynomial function of degree 4, which is shifted 4 units to the right and 4 units up.

Step 3 ：Since the domain is \(x \geq 4\), the function is increasing in this interval. Therefore, it has an inverse function.

Step 4 ：To find the inverse function, we replace \(g(x)\) with \(y\), swap \(x\) and \(y\), and then solve for \(y\).

Step 5 ：The equation becomes \(x = (y - 4)^4 + 4\).

Step 6 ：Solving for \(y\), we get \(y = 4 - \sqrt[4]{x - 4}\).

Step 7 ：However, this function is not defined for \(x < 4\), which is consistent with the original domain of \(x \geq 4\).

Step 8 ：Thus, the inverse function is \(y = 4 - \sqrt[4]{x - 4}\) for \(x \geq 4\).

Step 9 ：\(\boxed{g^{-1}(x)= 4 - \sqrt[4]{x - 4}, x \geq 4}\)