Let's break this down step by step:

1. Revenue function: Revenue is the total income the firm makes from selling products. If the firm sells each unit for $50, and $x$ represents the number of units sold, the revenue function $R(x)$ is:

   $R(x) = 50x$

   This is because the firm earns $50 for each of the $x$ units sold.

2. Cost function: The cost is the total expense incurred for producing and selling $x$ units. We are given that the cost for production and sales is $200,000 per unit. So, the cost function $C(x)$ is:

   $C(x) = 200,000x$

3. Profit function: Profit is the difference between the revenue and the cost. Therefore, the profit function $P(x)$ is:

   $P(x) = R(x) - C(x) = 50x - 200,000x = -199,950x$

   This shows that for each unit sold, the firm is losing $199,950 in profit, meaning the firm operates at a loss.

Sketching the functions

• Revenue function $R(x) = 50x$: This is a straight line with a positive slope of 50, passing through the origin. The graph will rise as more units are sold.

• Cost function $C(x) = 200,000x$: This is also a straight line, but with a much steeper slope of 200,000, passing through the origin. The cost grows much faster than revenue as production increases.

• Profit function $P(x) = -199,950x$: This is a straight line with a negative slope, showing a loss that increases linearly as more units are produced and sold.

I will now provide a sketch of these functions for clarity.

Want to see the sketch?

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Related Questions:

1. How would the profit function change if the cost per unit decreases?
2. What happens to the break-even point if the revenue per unit increases to $200?
3. How would a fixed cost (e.g., a $50,000 setup cost) impact the cost and profit functions?
4. What if the revenue per unit starts declining after producing a certain number of units?
5. How would the graph look if the cost was a quadratic function due to increasing costs for higher production?

Tip: The slope of the profit function indicates how profit changes with each additional unit produced. A negative slope means a loss per unit.