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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 functions1
Constructed Functions
  • Functions can be used to create new functions
    • Addition/subtraction
    • Multiplication/division
    • Composition
    • Piecewise connection
    • Inverting
constructed functions2
Constructed Functions
  • Revenue
    • Proceeds from sales
  • Costs
    • Expenditures from operations

Profit = Revenue - Cost

Profit = r(t) - c(t)

p(t) = r(t) - c(t)

constructed functions3
Constructed Functions

Revenue

Cost

Profit

constructed functions4
Constructed Functions

Output

Price

Revenue

= Price x Output

constructed functions6
Constructed Functions
  • Function composition
    • Output of one function is the input to another function
constructed functions7
Constructed Functions

Adv. Dollars

Adv. Dollars

Hours

Rule f

Rule g

Hours

Sales

Sales

constructed functions8
Constructed Functions

Adv. Dollars

Rule f

Hours

Rule g

Sales

constructed functions9
Constructed Functions

Adv. Dollars

Rule h

Sales

constructed functions10
Constructed Functions

Hours from dollars

Sales from hours

Sales from dollars

constructed functions11
Constructed Functions

Output from workers

Sales from output

Sales from workers

constructed functions13
Constructed Functions
  • Piecewise Functions
    • Form of the function changes at certain input values
    • Input values are known as break points
constructed functions14
Constructed Functions
  • Ex: Volume discounts for steel purchases
constructed functions17
Constructed Functions
  • Ex: Price rebates
constructed functions21
Constructed Functions
  • Inverse Functions
    • Reverses the input and the output
    • Preserves the relationship between them
constructed functions22
Constructed Functions

Adv. Hours

Sales

Rule f-1

Rule f

Sales

Adv. Hours

constructed functions24
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
slide26

Sales

Hours

slide27

Hours

Sales

slide28

Sales

Hours

constructed functions25
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
constructed functions29
Constructed Functions
  • Ex: Advertising hours to sales
constructed functions30
Constructed Functions
  • Ex: Profit (y) from sales (z)
constructed functions32
Constructed Functions
  • Composition of Inverse Functions
    • A function and it’s inverse “cancel each other out”