Section 2'4 Composite and Inverse Functions

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Section 2.4Composite and Inverse Functions
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• Consider the following situation
• There is an oil spill and the oil is
  disseminating from the source in a circular
  fashion
• We take some measurements and notice that after
  one second a particular point at the edge of the
  slick is 5 feet from the source, at 2 seconds
  that same point is 10 feet from the source, at 3
  seconds, it is 15 feet from the source
• What we need is to develop a function that will
  give us the area of the oil slick in terms of
  time
• What formulas may be useful here?
• Come up with a formula for the area of the slick
  and the point seperately

3

• When the output of one function is the input for
  another function, the two functions form a
  composition
• In the previous example the function giving the
  radius as a function of time became the input for
  the function that gave the area as a function of
  the radius
• We could write A f(r(t)) g(t)
• We now have that area is a function of time
• We essentially cut out the middle man

4

• The following table gives the population, P, of
  the town of Jonesville as a function of time, t,
  where t is the number of years since 1980
• Come up with a formula for P f(t)
• Tell in what year the population reaches 30,000
• Can we always find the year given the population?

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• Now if we swap the independent and dependent
  variable, we get the following table
• Do we still have a function?
• Find the inverse function, f -1(P) for this
  problem
• Lets see what happens when we compose f and f -1
• Turns out if f and are inverses we have

6

• The revenue, R, in thousands of dollars for
  selling x units of a given product is given by R
  f(x) 500x
• Find and interpret
• f(10)
• f -1(1000)
• f -1(R)
• In your groups work on problems 1, 7, and 17