Consider two cases in which cart a moves to the left with momentum \vec{p}_ip → i and then collides with a second cart which is initially at rest on a horizontal, frictionless air track. in case 1, cart a moves to the left after the collision at a speed of 1 m/s. in case 2, cart a moves to the right after the collision with the same speed as in case 1 (1 m/s). after the collision, the magnitude of the second cart's momentum in case 1 is the magnitude of the second cart's momentum in case 2.

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we have been given a quadratic function and we need to restrict the domain such that it becomes a one to one function.

we know that vertex of this quadratic function occurs at (5,2).

further, we know that range of this function is .

if we restrict the domain of this function to either or , it will become one to one function.

let us know find its inverse.

upon interchanging x and y, we get:

let us now solve this function for y.

hence, the inverse function would be if we restrict the domain of original function to and the inverse function would be if we restrict the domain to .