Determine the inverse of the function f(x):
Let
\[
\begin{array}{l}
f(x)=3-x^{2}, x \geq 0 \\
f^{-1}(x)=
\end{array}
\]

Answer

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Answer

\(\boxed{f^{-1}(x) = \sqrt{3 - x}}\) is the final answer.

Steps

Step 1 ：Given the function \(f(x) = 3 - x^2\), we can write it as \(y = 3 - x^2\).

Step 2 ：Switch x and y to get \(x = 3 - y^2\).

Step 3 ：Isolate \(y^2\) to get \(y^2 = 3 - x\).

Step 4 ：Take the square root of both sides to get \(y = \sqrt{3 - x}\).

Step 5 ：So, the inverse function \(f^{-1}(x)\) is \(f^{-1}(x) = \sqrt{3 - x}\).

Step 6 ：Check the work by substituting \(f^{-1}(x)\) into the original function to get \(f(f^{-1}(x)) = 3 - (\sqrt{3 - x})^2 = 3 - (3 - x) = x\).

Step 7 ：\(\boxed{f^{-1}(x) = \sqrt{3 - x}}\) is the final answer.