## Problem Restatement **Given:** \( f(x) = \sqrt{x+1} \) for \( \geq1 \) **:** Find inverse function \(^{-1}() \) ( the given interval), and explain each step. --- ## Step 1: Set \( y = f(x) \) and Solve for \( x \) Set \( y = \sqrt{x+1} \). **Solve for \( x \):** \[ y = \sqrt{x+1} \] Square both sides to remove the square root: \[ y^2 = x + 1 \] Now isolate \( x \): \[ x = y^2 - 1 \] **Explanation:** To find the inverse, we swap the roles of \( x \) and \( y \), and solve for \( x \) in terms of \( y \). --- ## Step 2: Express the Inverse Function Now, replace \( y \) with \( x \) to write the inverse function: \[ f^{-1}(x) = x^2 - 1 \] **Domain of the Inverse:** Since the original function \( f(x) \) is defined for \( x \geq -1 \), \( f(x) \) outputs values \( y \geq \). So, the inverse function's domain is \( x \geq \). **Explanation:** We substitute \( y \) with \( x \) to write the inverse function, and adjust the domain according to the range of the original function. --- ## Final Answer \[ \boxed{ f^{-1}(x) = x^2 - 1 \quad \text{for} \quad x \geq } \]