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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 3Transforming Functions

3.1 Transformations

3.2 Sequential Relationships

3.3 Inverse Functions

More functions (beyond linear and exponential)

More complicated functions

slide3

3.2 Sequential Relationships

  • In Context
  • Algebraically
slide4

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.

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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?

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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!

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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

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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

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3.3 Inverse Functions

  • Algebraically
  • Graphically
  • In Context
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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).

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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).

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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.

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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!

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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)

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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)

slide17

Homework:

Pages 131-133: #15-32

Turn In: 17a, 18c, 24, 26, 28, 30