Find the inverse of this function. $f(x)=\frac{4}{x+7}$
$f^{-1}(x)=-\frac{4}{x}+7$
$f^{-1}(x)=\frac{-7}{x}-4$
$f^{-1}(x)=\frac{4}{x}-7$
$f^{-1}(x)=\frac{\sqrt{4}}{-x}-7$

Step 1 ：The problem is asking for the inverse of the function \(f(x)=\frac{4}{x+7}\).

Step 2 ：To find the inverse of a function, we need to switch the roles of \(x\) and \(y\) and solve for \(y\).

Step 3 ：Let's start by replacing \(f(x)\) with \(y\), so we have \(y=\frac{4}{x+7}\).

Step 4 ：Next, we swap \(x\) and \(y\) to get \(x=\frac{4}{y+7}\).

Step 5 ：Now, we solve for \(y\) to get the inverse function. We can do this by multiplying both sides by \(y+7\) and then subtracting 7 from both sides.

Step 6 ：This gives us the inverse function \(f^{-1}(x)=-7 + \frac{4}{x}\).

Step 7 ：\(\boxed{f^{-1}(x)=-7 + \frac{4}{x}}\) is the final answer.