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Chapter 3 Transforming Functions

Chapter 3 Transforming Functions. Chapter 3 Transforming Functions. 3.1 Transformations 3.2 Sequential Relationships 3.3 Inverse Functions. More functions (beyond linear and exponential) More complicated functions. 3.2 Sequential Relationships. In Context Algebraically.

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Chapter 3 Transforming Functions

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  1. Chapter 3Transforming Functions

  2. Chapter 3Transforming Functions 3.1 Transformations 3.2 Sequential Relationships 3.3 Inverse Functions More functions (beyond linear and exponential) More complicated functions

  3. 3.2 Sequential Relationships • In Context • Algebraically

  4. Sequential RelationshipsIn Context Example: This week, an item of jewelry is on sale for 60% off. In addition, you have a coupon that will save you $10 on any jewelry purchase. Do you want to give the coupon to the clerk before or after she calculates the 60% reduction, or does it matter? Think Mathematically (in terms of functions!) COUPON: c(x) = x – 10 for x the price of the jewelry. DISCOUNT: d(x) = .40*x for x the price of the jewelry.

  5. COUPON then DISCOUNT x x-10 .40*(x-10) DISCOUNT then COUPON x .40*x .40*x - 10 Sequential RelationshipsIn Context Do you want to give the coupon to the clerk before or after she calculates the 60% reduction, or does it matter?

  6. Sequential RelationshipsIn Context COUPON: c(x) = x – 10DISCOUNT: d(x) = .40*xDISCOUNT after COUPON: f(x) = .40*(x – 10) COUPON after DISCOUNT: g(x) = .40*x -10 Notation: f(x) = d o c (x) = d(c(x)) g(x) = c o d (x) = c(d(x)) Composition of Functions ORDER is important rightmost (or inside) function FIRST!

  7. Sequential RelationshipsIn Context Example 4/111: A bank offers 1.12 euros for each American dollar. The rate to convert euros to Japanese yen is 108.93 yen to the euro. Name and write three functions: one to convert dollars to euros, one to convert euros to yen, and a third (their composition) to convert dollars to yen. D to E: e(x) = 1.12*x for x the amount of dollars. E to Y: y(x) = 108.93*x for x the amount of euros. x 1.12*x 108.93*(1.12*x) D to Y: f(x) = y o e(x) = y(e(x)) = 108.93*(1.12*x) input: dollars output: yen

  8. Sequential RelationshipsAlgebraically When functions g and then f are performed in sequence, the result f(g(x)) is called the composition of f with g and is denoted by the symbol f o g. The symbol f o g (x) indicates that the function g is performed first. More Practice: 1/113 and 2/114 and 3/111

  9. 3.3 Inverse Functions • Algebraically • Graphically • In Context

  10. Inverse FunctionsAlgebraically f(x) = 2*x – 1 and g(x) = (x+1)/2 What do you observe? If f(x) = y, then g(y) = x. If g(x) = y, then f(y) = x. ordered pair reversal ! What do you observe? g o f (x) = g(f(x)) = x f o g (x) = g(f(x)) = x. identity composition ! We say that g is the inverse function of f and write g(x) = f -1(x).

  11. Inverse FunctionsGraphically f(x) = 2*x – 1 and g(x) = (x+1)/2 symmetry about y = x We say that g is the inverse function of f and write g(x) = f -1(x).

  12. Inverse FunctionsProperties If an ordered pair (a,b) belongs to a function, then the ordered pair (b,a) belongs to its inverse. To obtain the graph of y = f -1(x), reflect the graph of y = f(x) about the line y = x. The heart of the relationship between f and its inverse function f -1(x) is this: f o f -1(x) = x for all x in the domain of f -1. f -1 o f (x) = x for all x in the domain of f.

  13. Inverse FunctionsIn Context Example p116: The conversion factor for changing US dollars to Mexican pesos is 9.2. The bank, however, first deducts a $15 service charge. f(x) = 9.2*(x-15) for x the amount of US dollars. If the traveler needs 5000 pesos, how many dollars must she exchange? 5000 = 9.2*(x-15) 5000/9.2 = x – 15 543.48 = x – 15 543.48 + 15 = x 558.48 = x. She must exchange 558.46 dollars. NOTE: Given output, we must find corresponding input!

  14. Inverse FunctionsIn Context Example p116: The conversion factor for changing US dollars to Mexican pesos is 9.2. The bank, however, first deducts a $15 service charge. f(x) = 9.2*(x-15) for x the amount of US dollars. If the traveler needs y pesos, how many dollars must she exchange? y = 9.2*(x-15) y/9.2 = x – 15 y/9.2 + 15 = x g(y) = y/9.2 + 15 The conversion function from pesos to dollars is given by: g(x) = x/9.2 + 15 for x the amount of pesos. NOTE: g(x) = f -1(x)

  15. Inverse FunctionsAlgebraically How to Find the Inverse of a Function (p121) • Rewrite the function using the independent-dependent variable notation, that is using x as the input and y as the output. • Solve the resulting equation for x. • Interchange the x and y variables. • Rename y as f -1(x) Find the inverse functions: 31/133, 25/132, 2/121, 2/123 (graph)

  16. 2/123 (graph)

  17. Homework: Pages 131-133: #15-32 Turn In: 17a, 18c, 24, 26, 28, 30

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