Constructed Functions

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
Hours
Adv. Dollars
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

Volume Price/ton
0 to 50 100
50 to 100 80
100 to 150 75
150 and up 50
16
Constructed Functions
17
Constructed Functions
18
Constructed Functions

• Ex Price rebates

Volume Rebate Per Unit
0 to 10 1000 5000
10 to 20 2000 5000
20 to 30 4000 5000
30 and up 8000 5000
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
(No Transcript)
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 its inverse cancel each other
  out

37
Constructed Functions

• f(x) x2

38
Constructed Functions

• f-1(x) x1/2

39
Constructed Functions

• In-Class