\( 1 \leftarrow \) The function \( f(x)=x+11 \) is one-to-one. a. Find an equation for \( f^{-1}(x) \), the inverse function. b. Verify that your equation is correct by showing that \( f\left(f^{-1}(x)\right)=x \) and \( f^{-1}(f(x))=x \). a. Select the correct choice below and fill in the answer box(es) to complete your choice. (Simplify your answer. Use integers or fractions for any numbers in the expression.) A. \( f^{-1}(x)=\square \), for \( x \geq \) B. \( f^{-1}(x)=\square \), for \( x \leq \square \) C. \( f^{-1}(x)=\square \), for \( x \neq \square \) D. \( f^{-1}(x)=\square \), for all \( x \)

To find the inverse function \( f^{-1}(x) \), we start with the original function \( f(x) = x + 11 \). To find the inverse, we switch \( x \) and \( y \) (where \( y = f(x) \)), giving us \( x = y + 11 \). Solving for \( y \), we get \( y = x - 11 \). Thus, the inverse function is \( f^{-1}(x) = x - 11 \), which applies for all \( x \). For verification, we check both conditions: 1. Calculating \( f(f^{-1}(x)) \): \( f(f^{-1}(x)) = f(x - 11) = (x - 11) + 11 = x \). 2. Now calculating \( f^{-1}(f(x)) \): \( f^{-1}(f(x)) = f^{-1}(x + 11) = (x + 11) - 11 = x \). Both conditions are satisfied, confirming that \( f^{-1}(x) = x - 11 \) is indeed the correct inverse function. Thus, the correct choice is: D. \( f^{-1}(x) = x - 11 \), for all \( x \)