Constructed Functions

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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 Functions

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Constructed Functions

Section 1.2

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

Revenue

Cost

Profit

Constructed Functions

Output

Price

Revenue

= Price x Output

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

Hours

Rule f

Rule g

Hours

Sales

Sales

Constructed Functions

Rule f

Hours

Rule g

Sales

Constructed Functions

Rule h

Sales

Constructed Functions

Hours from dollars

Sales from hours

Sales from dollars

Constructed Functions

Output from workers

Sales from output

Sales from workers

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

Sales

Rule f-1

Rule f

Sales

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

Sales

Hours

Hours

Sales

Sales

Hours

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