The Reciprocal Function and its Inverse

Use the sliders to change the coefficients and constant in the reciprocal function. The blue graph is the function; the red graph is its inverse. Try to find functions that are self-inverse, i.e. the red graph and blue graph will be the same. When you find one, make a note of the values of a, b, c and d. Try to find more examples, recording the values on the sliders for each one.
What is the condition for a reciprocal function to be self-inverse? Prove it!