The cost in dollars to produce x t-shirts is C(x)=2.5 x+50 ....

The cost in dollars to produce $x$ t-shirts is $C(x)=2.5 x+50 .$ The revenue in dollars from sales of $x$ t-shirts is $R(x)=10.99 x$.
(a) Write and simplify a function $P$ that gives profit in terms of $x$.
(b) Find the profit if $100 \mathrm{t}$ -shirts are produced and sold.

Key Concepts

Function Evaluation

Function evaluation involves substituting a specific value into a function to calculate its output. This concept is essential for determining precise outcomes, such as the total profit when a known number of items is produced and sold, making it vital for practical applications in business and economics.

Revenue Function

A revenue function calculates the total income generated from selling a certain number of items. It is typically a product of the number of units sold and the price per unit. Understanding the revenue function is important for evaluating the income potential of selling products or services and for assessing overall business performance.

Profit Function

The profit function is derived by subtracting the cost function from the revenue function, thereby representing the net financial gain or loss from production and sales. This function is central to business analysis and decision-making because it directly indicates the economic viability of producing and selling a product.

Linear Functions

Linear functions are mathematical models that depict a relationship with a constant rate of change and are typically expressed in the form f(x) = mx + b. They are highly beneficial in business contexts for modeling costs, revenues, and profits, as they provide a straightforward way to understand how a small change in one quantity leads to a consistent change in another.

Cost Function

A cost function represents the total expenditure incurred in the production of a certain number of items. It usually consists of a variable cost component, which depends on the number of units produced, and a fixed cost component, which remains the same regardless of the production volume. This function is crucial for determining how production costs scale with changes in output.

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