Example Question - finding inverse

Finding the Inverse Function of a Given Function

To find \( f^{-1}(x) \), the inverse of the function \( f(x) = \sqrt{x} - 10 \), follow these steps: 1. Replace \( f(x) \) with \( y \): \[ y = \sqrt{x} - 10 \] 2. Swap \( x \) and \( y \) to find the inverse: \[ x = \sqrt{y} - 10 \] 3. Solve for \( y \): \[ x + 10 = \sqrt{y} \] Now square both sides to get rid of the square root: \[ (x + 10)^2 = (\sqrt{y})^2 \] \[ x^2 + 20x + 100 = y \] The inverse function \( f^{-1}(x) \) is then: \[ f^{-1}(x) = x^2 + 20x + 100 \]