Given the function \( f(x)=\sqrt{x-2} \), complete parts a through \( c \). (a) Find an equation for \( f^{-1}(x) \). (b) Graph \( f \) and \( f^{-1} \) in the same rectangular coordinate system. (c) Use interval notation to give the domain and the range of \( f \) and \( f^{-1} \). (Hint: To solve for a variable involving an nth root, raise both sides of the equation to the nth power, \( (\sqrt[n]{y})^{n}=y \).) a) Find \( f^{-1}(x) \). 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.) B. \( f^{-1}(x)=\square, x \neq \square \) C. \( f^{-1}(x)=\square, x \leq \square \) D. for all \( x \) D. \( f^{-1}(x)=\square, x \geq \square \)

We are given f(x) = √(x – 2). Step 1. Find f⁻¹(x): 1. Write y = √(x – 2). 2. To solve for x in terms of y, square both sides:   y² = x – 2. 3. Add 2 to both sides:   x = y² + 2. 4. Now, swap the roles of x and y to get the inverse function:   f⁻¹(x) = x² + 2. Step 2. Determine the proper domain for the inverse: Since f(x) = √(x – 2) has a domain of x ≥ 2 and a range of y ≥ 0, the inverse function f⁻¹(x) has:  • Domain: x ≥ 0 (the range of f)  • Range: y ≥ 2 (the domain of f) Step 3. Graphing the functions: The graph of f(x) = √(x – 2) is a horizontal shift of the basic square root graph (shifted right by 2). It starts at the point (2, 0) and increases slowly. The graph of f⁻¹(x) = x² + 2 is a vertical parabola opening upward with its vertex at (0, 2). These two graphs are reflections of each other across the line y = x. Step 4. Answering the parts: (a) The correct choice is:   D. f⁻¹(x) = x² + 2, x ≥ 0 (b) To graph f and f⁻¹ in the same rectangular coordinate system:  – Plot f(x) = √(x – 2) starting at (2, 0) for x ≥ 2.  – Plot f⁻¹(x) = x² + 2 for x ≥ 0 (so starting at (0, 2)).  – Note that they are mirror images about the line y = x. (c) Domains and Ranges:  For f(x) = √(x – 2):   • Domain: [2, ∞)   • Range: [0, ∞)  For f⁻¹(x) = x² + 2:   • Domain: [0, ∞)   • Range: [2, ∞) Thus, the final answer for part (a) is:  D. f⁻¹(x) = x² + 2, x ≥ 0.