How is the work done to hoist the counterweight related to the potential energy of the counterweight at its specified height?

The work done to lift the counterweight equals the potential energy acquired

Explanation:

since this is vertically applied force on the counterweight, and the distance the force is displacing the counterweight is in the same direction as the applied force, it equals the gained potential energy

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 .