Restrict the domain of the function f(x) = (x-2)^2 so it has an inverse. Then determine its inverse function.

Answer:Look to the bold answer downStep-by-step explanation:* Lets explain how to restrict the domain of the quadratic function- The quadratic function is two-to-one function - The inverse of it is one-to-two which is not a function- So we can not find the inverse of the quadratic function until restrict its  domain- We restrict the domain at the x-coordinate of the vertex of the function∵ f(x) = (x - h)² + k is the standard form of the quadratic function, where   (h , k) are the coordinates of its vertex- To restrict the domain we put x > h for the right part of the parabola   or x < h for the left part of the parabola* Lets solve the problem∵ f(x) = (x - 2)²∵ f(x) = (x - h)² + k is the standard form of the quadratic function∴ h = 2 and k = 0∴ The vertex of the parabola is (2 , 0)- We will restrict the domain at x = 2∴ The domain of the function f(x) to have inverse is x > 2 or x < 2* The restriction domain is x > 2 or x < 2- To find the inverse of the function switch x and y and solve for the   new y∵ f(x) = (x - 2)²∵ f(x) = y∴ y = (x - 2)²- Switch x and y∴ x = (y - 2)² - take square root for both sides∴ ± √x = y - 2 - Add 2 for both sides∴ ± √x + 2 = y∴ [tex]f^{-1}=\sqrt{x}+2=====OR=====f^{-1}=-\sqrt{x}+2[/tex]* For the domain x > 2 of f(x) the inverse is [tex]f^{-1}(x) = \sqrt{x}+2[/tex]  For the domain x < 2 of f(x) the inverse is [tex]f^{-1}(x)=-\sqrt{x}+2[/tex]