Inverse Function F9Ac8C

1. **State the problem:** We are given the function $f(x) = \sqrt{4 - x}$ and need to find its inverse function $f^{-1}(x)$. 2. **Recall the definition of inverse function:** The inverse function $f^{-1}(x)$ satisfies $f(f^{-1}(x)) = x$ and $f^{-1}(f(x)) = x$. To find $f^{-1}(x)$, we start by setting $y = f(x)$, so $y = \sqrt{4 - x}$. 3. **Express $x$ in terms of $y$:** Square both sides to remove the square root: $$y^2 = 4 - x$$ Rearrange to solve for $x$: $$x = 4 - y^2$$ 4. **Swap variables to get inverse function:** Replace $y$ by $x$ and $x$ by $f^{-1}(x)$: $$f^{-1}(x) = 4 - x^2$$ 5. **Check domain and range:** Since $f(x) = \sqrt{4 - x}$, the domain of $f$ is $x \leq 4$ and the range is $y \geq 0$. The inverse function swaps domain and range, so $f^{-1}(x)$ has domain $x \geq 0$ and range $f^{-1}(x) \leq 4$. This matches the formula $f^{-1}(x) = 4 - x^2$. **Final answer:** $$f^{-1}(x) = 4 - x^2$$