Composite and Inverse Functions

1
Composite and Inverse Functions

• Lesson 2.4

2
Composition of Functions

• Consider two functions where the output of one
  is the input of the next
• Example
• Square yds/hr mowed is a function of how fast
  you push the mowerA f(s)
• The time required to mow is a function of square
  yds/hr you cover T g(A)

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Composition of Functions

• Given the following functions
• Q f(p) The number of barrels of oil sold when
  the price is p dollars per barrel
• R(Q) is the revenue earned when Q barrels are
  sold
• What is R(f(p)) ?
• What are the units of each function?

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Composition of Functions

• Given
• Find the following compositions

Try using your calculator
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Inverse Functions

• What if we cram a numberup the spout of a
  function and out of the funnel popsthe number
  that wouldhave given us the result??
• The function that does this is called theinverse
  function

Use spreadsheet to evaluate inverse of a function
6
Perspectives for Input and Output

• Suppose you are told 1 gallon of paint covers
  250 ft2
• You might derive the function
• It is just as reasonable to consider how many
  gallons are needed for a certain area

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Perspectives for Input and Output

• The mathematical relationship is the same
• The input on one f(g) is the output on h(A)
• We would say the functions have an inverse
  relationship

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Inverse Function Notation

• For the inverse of function f, we use the
  notation f -1
• Note that this is not the same as a negative
  exponent
• It is not

9
Finding Inverse Values from a Table

• Given the following table which defines the
  function f
• Determine
• f(-2)
• f -1(2)
• f -1(-4)
• f(-1)

x -2 -1 0 1 2
f(x) 6 -4 3 2 9
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Finding Inverse Values from a Graph

• Write some ordered pairsfor the
  functiondefined by thisgraph
• Determinef -1(0)f -1(-2)

x f(x)

• Are there multiple answers
• Is the inverse even a function?

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Finding the Inverse Formula

• Given the formula
• Find the inverse function f -1(V)
• Strategy
• Write in formula notation
• Solve for the independent variable r ?

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Domain and Range of An Inverse Function

• Note that the domain of the original function
  becomes the range of the inverse
• Thus restrictions on the original domain affect
  the range of the inverse
• AlsoThe range of the original may be restricted
• This affects the domain of the inverse
• Consider the inverses of these functions

As we saw on slide 10, some inverses might not
even be functions
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Assignment

• Lesson 2.4
• Page 82
• Exercises1 37 odd