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The cost function represents the total cost incurred in producing a certain number of items, which in this case is cell phones. This function considers both variable and fixed costs. In our example, the cost function is given by \(C(x) = 150x + 10,000\). Here, the term \(150x\) represents the variable cost, which changes with the number of units produced — in this case, each unit produced incurs an additional $150 cost. Meanwhile, \(10,000\) is the fixed cost, which is the cost of production that does not change, regardless of the number of units produced. This could include expenses like rent, salaries, and utility bills that organize the production facilities. Understanding the breakdown in a cost function is crucial for determining profitability and making informed business decisions.

The revenue function defines the total income a company earns from selling its products. It is a straightforward calculation of the number of items sold multiplied by the price of each item. In our scenario, the revenue function is \(R(x) = 200x\). This implies that each unit sold brings in \(200. Therefore, for every phone sold, the company gains \)200 in revenue, directly correlating their income to their sales volume. Analyzing the revenue function helps businesses predict income levels based on expected sales volumes. It is critical when planning strategies to reach break-even points and potential profitability.

Algebraic equations involve finding unknown variables within mathematical expressions by organizing and simplifying the data represented. They are incredibly useful in many real-world problems, including financial calculations like determining break-even points.In the exercise, the break-even point is found by setting the cost function equal to the revenue function, forming the equation \(150x + 10,000 = 200x\). Solving this equation involves basic algebraic manipulation:

• Starting with setting the cost equal to the revenue to identify where no profit or loss is made.
• Isolating the variable involves manipulating both sides to get terms involving \(x\) on one side of the equation, resulting in the equation \(50x = 10,000\).
• Finally, solving for \(x\) by dividing both sides by 50 gives us \(x = 200\).

This solution signifies that when 200 phones are produced and sold, the total production cost equals the revenue, reaching the break-even point where no net gain or loss occurs.