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Composite and Inverse Functions

Understand & utilize composite & inverse functions, explore relationships, find values from tables & graphs. See examples & practice exercises for a solid grasp.

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Composite and Inverse Functions

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  1. Composite and Inverse Functions Lesson 2.4

  2. 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)

  3. 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?

  4. Composition of Functions • Given • Find the following compositions Try using your calculator

  5. 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

  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

  7. 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

  8. 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)

  10. 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?

  11. 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 = ?

  12. 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

  13. Assignment • Lesson 2.4 • Page 82 • Exercises1 – 37 odd

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