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Understanding Constructed Functions: Revenue, Costs, and Profit

This guide explores constructed functions in mathematics, focusing on their application in revenue, costs, and profit calculations. It covers how to create new functions through operations such as addition, subtraction, multiplication, and division. In particular, we will delve into function composition, piecewise functions, and the importance of one-to-one functions for finding inverses. Examples will illustrate concepts such as revenue from sales, costs of expenditures, and calculating profit. Enhance your understanding of how constructed functions play a crucial role in economic analysis.

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Understanding Constructed Functions: Revenue, Costs, and Profit

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  1. Constructed Functions Section 1.2

  2. Constructed Functions • Functions can be used to create new functions • Addition/subtraction • Multiplication/division • Composition • Piecewise connection • Inverting

  3. Constructed Functions • Revenue • Proceeds from sales • Costs • Expenditures from operations Profit = Revenue - Cost Profit = r(t) - c(t) p(t) = r(t) - c(t)

  4. Constructed Functions Revenue Cost Profit

  5. Constructed Functions Output Price Revenue = Price x Output

  6. Constructed Functions • In-Class

  7. Constructed Functions • Function composition • Output of one function is the input to another function

  8. Constructed Functions Adv. Dollars Adv. Dollars Hours Rule f Rule g Hours Sales Sales

  9. Constructed Functions Adv. Dollars Rule f Hours Rule g Sales

  10. Constructed Functions Adv. Dollars Rule h Sales

  11. Constructed Functions Hours from dollars Sales from hours Sales from dollars

  12. Constructed Functions Output from workers Sales from output Sales from workers

  13. Constructed Functions • In-Class

  14. Constructed Functions • Piecewise Functions • Form of the function changes at certain input values • Input values are known as break points

  15. Constructed Functions • Ex: Volume discounts for steel purchases

  16. Constructed Functions

  17. Constructed Functions

  18. Constructed Functions • Ex: Price rebates

  19. Constructed Functions

  20. Constructed Functions

  21. Constructed Functions • In-Class

  22. Constructed Functions • Inverse Functions • Reverses the input and the output • Preserves the relationship between them

  23. Constructed Functions Adv. Hours Sales Rule f-1 Rule f Sales Adv. Hours

  24. Constructed Functions

  25. Constructed Functions • One-to-One Function • For an inverse to exist, the original function must be a one-to-one function • If for any two different inputs, you get two different outputs, then the function is a one-to-one function

  26. Sales Hours

  27. Hours Sales

  28. Sales Hours

  29. Constructed Functions • Horizontal line test • If at any output, a horizontal line crosses at more than one point, that function does not have an inverse

  30. Constructed Functions Yes

  31. Constructed Functions No

  32. Constructed Functions Yes

  33. Constructed Functions • Ex: Advertising hours to sales

  34. Constructed Functions • Ex: Profit (y) from sales (z)

  35. Constructed Functions • Rules

  36. Constructed Functions • Composition of Inverse Functions • A function and it’s inverse “cancel each other out”

  37. Constructed Functions • f(x) = x2

  38. Constructed Functions • f-1(x) = x1/2

  39. Constructed Functions • In-Class

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