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Constructed Functions

Constructed Functions. Section 1.2. Constructed Functions. Functions can be used to create new functions Addition/subtraction Multiplication/division Composition Piecewise connection Inverting. Constructed Functions. Revenue Proceeds from sales Costs Expenditures from operations.

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Constructed Functions

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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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