Total revenue and profit are two important concepts in Business Mathematics and Statistics because they help a firm determine its sales performance and financial position. Revenue represents the income earned by a firm from selling its goods or services, while profit represents the excess of total revenue over total cost. Mathematical functions can be used to derive the total revenue and profit of a firm from its demand, price, output, and cost relationships. Let the quantity of output or units sold by the firm be represented by (Q), and let the price per unit be represented by (P). The total revenue of the firm is the total amount received from selling (Q) units at price (P). Therefore, [ TR = P \times Q ] Thus, total revenue is obtained by multiplying the price per unit by the quantity sold. If the price remains constant, say (P=a), then total revenue is [ TR=aQ ] This represents a straight-line total revenue function passing through the origin. For example, if a firm sells 100 units at ₹50 per unit, then [ TR=50\times100=₹5,000 ] However, in many practical situations, price does not remain constant. A firm may have to reduce its price to sell a larger quantity. In such a case, price becomes a function of quantity demanded. Suppose the demand or price function is [ P=a-bQ ] where (a) is the maximum possible price and (b) represents the rate at which price decreases as output increases. Substituting this price function into the total revenue formula gives [ TR=PQ ] [ TR=(a-bQ)Q ] Therefore, [ \boxed{TR=aQ-bQ^2} ] This is the total revenue function of the firm when price depends linearly on quantity. The total revenue function can also be derived from the demand function. Suppose the demand function is [ Q=\frac{a-P}{b} ] Rearranging, [ bQ=a-P ] and hence, [ P=a-bQ ]