Composite and Inverse Functions

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Presentation transcript:

Composite and Inverse Functions Lesson 2.4

Composition of Functions speed sq yds/hr f(s) 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) The time required to mow is a function of square yds/hr you cover T = g(A) Time g(A)

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 Functions Given Find the following compositions Try using your calculator

Use spreadsheet to evaluate inverse of a function Inverse Functions What if we cram a number up the spout of a function and out of the funnel pops the number that would have given us the result?? The function that does this is called the inverse function 1 -3 4 3 3 1 Use spreadsheet to evaluate inverse of a function

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 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 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 Given the following table which defines the function f Determine f(-2) f -1(2) f -1(-4) f(-1) x -2 -1 1 2 f(x) 6 -4 3 9

Finding Inverse Values from a Graph Write some ordered pairs for the function defined by this graph Determine f -1(0) f -1(-2) x f(x) Are there multiple answers Is the inverse even a function?

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 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 Also The 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 Lesson 2.4 Page 90 Exercises 1 – 37 Odd