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Composite and Inverse Functions. Lesson 2.4. speed. f(s). sq yds/hr. g(A). Time. 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 mower A = f(s)

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composition of functions

speed

f(s)

sq yds/hr

g(A)

Time

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)
composition of functions1
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?
composition of functions2
Composition of Functions
  • Given
  • Find the following compositions

Try using your calculator

inverse functions

1

4

3

-3

3

1

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

perspectives for input and output
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
perspectives for input and output1
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
inverse function notation
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
finding inverse values from a table
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)
finding inverse values from a graph
Finding Inverse Values from a Graph
  • Write some ordered pairsfor the functiondefined by thisgraph
  • Determinef -1(0)f -1(-2)
  • Are there multiple answers
  • Is the inverse even a function?
finding the inverse formula
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 = ?
domain and range of an inverse function
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

assignment
Assignment
  • Lesson 2.4
  • Page 82
  • Exercises1 – 37 odd
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