17
5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washin

1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

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Page 1: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

1.5 Functions and Logarithms

Greg Kelly, Hanford High School, Richland, Washington

Page 2: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

A relation is a function if:for each x there is one and only one y.

A relation is a one-to-one if also: for each y there is one and only one x.

In other words, a function is one-to-one on domain D if:

f a f b whenever a b

Page 3: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

To be one-to-one, a function must pass the horizontal line test as well as the vertical line test.

31

2y x 21

2y x 2x y

one-to-one not one-to-one not a function

(also not one-to-one)

Page 4: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

Inverse functions:

11

2f x x Given an x value, we can find a y value.

11

2y x

11

2y x

2 2y x

2 2x y

Switch x and y: 2 2y x 1 2 2f x x

(eff inverse of x)

Inverse functions are reflections about y = x.

Solve for x:

Page 5: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

example 3: 2f x x 0x

Graph: f x 1f x y x for 0x

a parametrically:

21 1: 0f x t y t t

1 22 2: f x t y t

3 3: y x x t y t

Y=

WINDOW

GRAPH

Page 6: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

GRAPH

WINDOW

example 3: 2f x x 0x

Graph: f x 1f x y x for 0x

b Find the inverse function:

21 0y x x Y=

2 x 0y x

y x

x y

Switch x & y:

y x1f x

Change the graphing mode to function.

>2y x

3y x

Page 7: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

Consider xf x a

This is a one-to-one function, therefore it has an inverse.

The inverse is called a logarithm function.

Example:416 2 24 log 16 Two raised to what power

is 16?

The most commonly used bases for logs are 10: 10log logx x

and e: log lne x x

lny x is called the natural log function.

logy x is called the common log function.

Page 8: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

lny x

logy x

is called the natural log function.

is called the common log function.

In calculus we will use natural logs exclusively.

We have to use natural logs:

Common logs will not work.

Page 9: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

Properties of Logarithms

loga xa x log xa a x 1 0a x

Since logs and exponentiation are inverse functions, they “un-do” each other.

Product rule: log log loga a axy x y

Quotient rule: log log loga a a

xx y

y

Power rule: log logya ax y x

Change of base formula:ln

loglna

xx

a

Page 10: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

Example 6:

$1000 is invested at 5.25 % interest compounded annually.How long will it take to reach $2500?

1000 1.0525 2500t

1.0525 2.5t We use logs when we have an

unknown exponent.

ln 1.0525 ln 2.5t

ln 1.0525 ln 2.5t

ln 2.5

ln 1.0525t 17.9 17.9 years

In real life you would have to wait 18 years.

Page 11: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

Example 7: Indonesian Oil Production (million barrels per year):

1960 20.56

1970 42.10

1990 70.10

Use the natural logarithm regression equation to estimate oil production in 1982 and 2000.

How do we know that a logarithmic equation is appropriate?

In real life, we would need more points or past experience.

Page 12: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

Indonesian Oil Production:

607090

20.56 million 42.10 70.10 60,70,90 L1 ENTER

2nd { 60,70,90 2nd } STO alpha L 1 ENTER

20.56,42.10,70.10 L2

LnReg L1, L2 ENTER

2nd MATH 6 3

Statistics Regressions

5

LnReg

alpha L 1 alpha L 2 ENTER

DoneThe calculator should return:

,

Page 13: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

ShowStat ENTER

2nd MATH 6 8

Statistics ShowStat

ENTER

The calculator gives you an equation and constants:

lny a b x 474.3

121.1

a

b

ExpReg L1, L2 ENTER

2nd MATH 6 3

Statistics Regressions

5

LnReg

alpha L 1 alpha L 2 ENTER

DoneThe calculator should return:

,

Page 14: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

We can use the calculator to plot the new curve along with the original points:

Y= y1=regeq(x)

2nd VAR-LINK regeq

x )

Plot 1 ENTER

ENTER

WINDOW

Page 15: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

Plot 1 ENTER

ENTER

WINDOW

GRAPH

Page 16: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

WINDOW

GRAPH

Page 17: 1.5 Functions and Logarithms Greg Kelly, Hanford High School, Richland, Washington

What does this equation predict for oil production in 1982 and 2000?

F3Trace

This lets us see values for the distinct points.

Moves to the line.

This lets us trace along the line.

82 ENTER Enters an x-value of 82.

100 ENTER Enters an x-value of 100.

In 1982, production was 59 million barrels.

In 2000, production was 84 million barrels.