Find the inverse function. \( f(x)=x^{2}+22 \) \( \left.f^{-1}(x)=\sqrt[{[?}]\right]{x+\square} \)

To find the inverse of the function \( f(x) = x^2 + 22 \), follow these steps: 1. **Replace \( f(x) \) with \( y \):** \[ y = x^2 + 22 \] 2. **Solve for \( x \) in terms of \( y \):** \[ y - 22 = x^2 \] \[ x = \pm\sqrt{y - 22} \] 3. **Define the inverse function:** Since a function must pass the vertical line test and have only one output for each input, we typically take the positive square root when defining the inverse (assuming the original function \( f(x) \) is restricted to \( x \geq 0 \)). \[ f^{-1}(x) = \sqrt{x - 22} \] So, the inverse function is: \[ f^{-1}(x) = \sqrt{x - 22} \] **Final Answer:** \( f^{-1}(x) = \sqrt{x\ -\ 22} \)