Consider the function \( f(x)=\sqrt{4-x} \). Find the formula for the inverse function.

To find the inverse of the function \( f(x) = \sqrt{4 - x} \), follow these steps: 1. **Start with the original equation:** \[ y = \sqrt{4 - x} \] 2. **Solve for \( x \) in terms of \( y \):** \[ y^2 = 4 - x \] \[ x = 4 - y^2 \] 3. **Express the inverse function:** \[ f^{-1}(y) = 4 - y^2 \] Typically, we replace \( y \) with \( x \) to express the inverse function in standard form: \[ f^{-1}(x) = 4 - x^2 \] **Domain and Range Considerations:** - **Original Function \( f(x) = \sqrt{4 - x} \):** - **Domain:** \( x \leq 4 \) (since the expression inside the square root must be non-negative) - **Range:** \( y \geq 0 \) (since square roots are non-negative) - **Inverse Function \( f^{-1}(x) = 4 - x^2 \):** - **Domain:** \( x \geq 0 \) (based on the range of the original function) - **Range:** \( y \leq 4 \) **Final Answer:** \[ f^{-1}(x) = 4 - x^2 \]