Complete the statements for the given function and its inverse. \[ f(x)=x^{2}+17, x \geq 0 \] The inverse function of \( f \) is \( f^{-1}(x)= \) The graph of \( f \) and the graph of \( f^{-1} \) are symmetric with respect to the The domain of \( f^{-1} \) is

Let's complete each part step by step for the function \( f(x) = x^{2} + 17 \) with \( x \geq 0 \). ### 1. Finding the Inverse Function \( f^{-1}(x) \): To find the inverse, solve for \( x \) in terms of \( y \): \[ y = x^{2} + 17 \] \[ y - 17 = x^{2} \] \[ x = \sqrt{y - 17} \] Since \( x \geq 0 \), we take the positive square root. Therefore, the inverse function is: \[ f^{-1}(x) = \sqrt{x - 17} \] ### 2. Symmetry of the Graphs: The graph of a function and its inverse are always symmetric with respect to the line: \[ y = x \] ### 3. Domain of the Inverse Function \( f^{-1}(x) \): For \( f^{-1}(x) = \sqrt{x - 17} \) to be real and defined: \[ x - 17 \geq 0 \] \[ x \geq 17 \] Thus, the domain of \( f^{-1} \) is: \[ x \geq 17 \] ### Summary: - **Inverse Function:** \( f^{-1}(x) = \sqrt{x - 17} \) - **Symmetry:** The graphs of \( f \) and \( f^{-1} \) are symmetric with respect to the line \( y = x \). - **Domain of \( f^{-1} \):** \( x \geq 17 \)