[1] Historically, the economists first expressed the price of a good as a function of demand (holding the other economic variables, like income, constant), and plotted the price-demand relationship with demand on the x (horizontal) axis (the demand curve).
Later the additional variables, like prices of other goods, came into analysis, and it became more convenient to express the demand as a multivariate function (the demand function):
[2] In mathematical terms, if the demand function is
[3] This is useful because economists typically place price (P) on the vertical axis and quantity (demand, Q) on the horizontal axis in supply-and-demand diagrams, so it is the inverse demand function that depicts the graphed demand curve in the way the reader expects to see.
[5] Note that although price is the dependent variable in the inverse demand function, it is still the case that the equation represents how the price determines the quantity demanded, not the reverse.
There is a close relationship between any inverse demand function for a linear demand equation and the marginal revenue function.
For any linear demand function with an inverse demand equation of the form P = a - bQ, the marginal revenue function has the form MR = a - 2bQ.
Total revenue equals price, P, times quantity, Q, or TR = P×Q.
This relationship holds true for all linear demand equations.
The importance of being able to quickly calculate MR is that the profit-maximizing condition for firms regardless of market structure is to produce where marginal revenue equals marginal cost (MC).
[11] Equating MR to MC and solving for Q gives Q = 20.
So 20 is the profit-maximizing quantity: to find the profit-maximizing price simply plug the value of Q into the inverse demand equation and solve for P.