Composite and inverse functions
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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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