Functions & Graphs (1.2)What is a function?Review Domain & RangeBoundednessOpen & Closed IntervalsDistance from a point to a line

Even & Odd Functions...

Ex: Identify the domain, range, (use interval notation) and whether the function is odd or even or neither.

y = x2

y = √(1-x2)

y = √x

y = 1/x

y = 2x/(x-1)

Functions Defined in Pieces

While some functions are defined by single formulas, others are defined by applying different formulas to different parts of their domain.

These are called piecewise functions.

Examples:

-x ; x < 0 y = x2 ; 0 < x < 1 1 ; x > 1

-x ; 0 < x < 1y =

2x – 2 ; 1< x < 2

The Absolute Value Function

The absolute value function is defined piecewise:

Composite Functions

Suppose that some of the outputs of a function can be used as inputs of

a function . We can then link and to form a new function whose inputs

are inputs of and whose outputs are the numbers

g

f g f

x g ( )( )( )( ) ( )

.

We say that the function read of of is

. The usual standard notation for the composite is ,

which is read " of ."

f g x

f g x f g x

f g

f g

the composite

of and og f

Examples

f(x) = x2 + 1 g(x) = x- 7

Find:g(f(2))

f(g(2))

g(g(3))

f(f(x))

g(f(x))

g(g(x))

Trig ReviewComplete Packet (will be part of

HW #6) on your ownSeek help either during seminar

or at next week’s review session

1.3Exponential Functions

Slide 1- 11

Exponential GrowthExponential DecayApplicationsThe Number e

…and why

Exponential functions model many growth patterns.

What you’ll learn about…

Slide 1- 12

Exponential Function

Let be a positive real number other than 1. The function

( )

is the .

x

a

f x a

a

=

exponential function with base

The domain of f(x) = ax is (-∞, ∞) and the range is (0, ∞). Compound interest investment and population growth are examples of exponential growth.

Slide 1- 13

Exponential Growth

If 1 the graph of looks like the graph

of 2 in Figure 1.22ax

a f

y=

Slide 1- 14

Exponential Decay

If 0 1 the graph of looks like the graph

of 2 in Figure 1.22b.x

a f

y -=

Slide 1- 15

Exponential Growth and Exponential Decay

The function , 0, is a model for

if 1, and a model for if 0 1.

xy k a k

a a

exponential growth

exponential decay

= × >

> < <

Graphing Exponential Functions

Graph y = 2x

◦x-intercept:_______

◦y -intercept:_______

◦Domain: _______◦Range: _______◦Type:

_______Slide 1- 16

Graph y = 2-x

x-intercept:_______

y -intercept:_______

Domain: _______ Range: _______ Type:

_______

Slide 1- 17

Rules for Exponents

See page 21 to review these!

Half-life

Exponential functions can also model phenomena that produce decrease over time, such as happens with radioactive decay. The half-life of a radioactive substance is the amount of time it takes for half of the substance to change from its original radioactive state to a non-radioactive state by emitting energy in the form of radiation.

Use the Law of Exponents to expand or condense

1. ax ay

2. (ax)y

3. ax bx

4. (a/b)y

Slide 1- 18

Slide 1- 19

Example Exponential Functions

( )Use a grapher to find the zero's of 4 3.xf x = -

( ) 4 3xf x = -

[-5, 5], [-10,10]

Rewrite the exponential expression to have the indicated base

(9)2x , base 3(1/8) 2x , base 2

Slide 1- 20

Applications The Population of Knoxville is 500,000 and is increasing at the rate

of 3.75% annually. Approximately when will the population reach 1 million?

Suppose the half-life of a certain radioactive substance is 20 days and that there are 5 g present initially. When will there only be 1 g of the substance left?

Interest ProblemsSimple Interest FormulaCompound Interest FormulaInterest compounded

continuously◦ How much would you get if P = $1, r =

100% and the principal were compounded continuously (every second of each day for 365 days) for one year?

Slide 1- 21

Slide 1- 22

The Number e

Many natural, physical and economic phenomena are best modeled

by an exponential function whose base is the famous number , which is

2.718281828 to nine decimal places.

We can define to be the numbe

e

e ( ) 1r that the function 1

approaches as approaches infinity.

x

f xx

x

æ ö÷ç= + ÷ç ÷çè ø

f(x) = (1 + 1/x)x

Slide 1- 23

The Number e

The exponential functions and are frequently used as models

of exponential growth or decay.

Interest compounded continuously uses the model , where is the

initial investment, is t

x x

r t

y e y e

y P e P

r

-= =

= ×

he interest rate as a decimal and is the time in years.t

Slide 1- 24

Example The Number e( ) 0.03

The approximate number of fruit flies in an experimental population after

hours is given by 20 , 0.

a. Find the initial number of fruit flies in the population.

b. How large is the populat

tt Q t e t= ³

ion of fruit flies after 72 hours?

c. Use a grapher to graph the function .Q

[0,100] by [0,120] in 10’s

Slide 1- 25

( )

( ) ( ) ( )

( ) ( )

0.03 0

0.03 2.72 16

0

a. To find the initial population, evaluate at 0.

20 20 20 1 20 flies.

b. After 72 hours, the population size is

20 2

0

0 173 flies.

c.

72

Q t t

Q e e

Q e e

=

= = = =

= = »

( ) 0.0320 , 0tQ t e t= ³

Slide 1- 26

Slide 1- 27

Quick Quiz Sections 1.1 – 1.3

( )

( )

( )

( )( )

You may use a graphing calculator to solve the following problems.

1. Which of the following gives an equation for the line through 3, 1

and parallel to the line: 2 1?

1 7A

2 21 5

B2 2

C 2 5

D 2

y x

y x

y x

y x

y x

-

=- +

= +

= -

=- +

=-

( )

7

E 2 1y x

-

=- +

Slide 1- 28

Quick Quiz Sections 1.1 – 1.3

( ) ( )( )( )

( )( )( )( )( )

22. If 1 and 2 1, which of the

following gives 2 ?

A 2

B 5

C 9

D 10

E 15

f x x g x x

f g

= + = -

o

Slide 1- 30

Warm-Up

( ) ( )( )

( )

In Exercises 1 3, write an equation for the line.

1. the line through the points 1, 8 and 4, 3

2. the horizontal line through the point 3, 4

3. the vertical line through the point 2, 3

In Exercises 4 - 6

-

-

-

2 2 2

2

, find the - and -intercepts of the graph of the relation.

4. 1 5. 19 16 16 9

6. 2 1

x y

x y x y

y x

+ = - =

= +

1.4Parametric Equations

Slide 1- 32

RelationsLines and Other Curves

What you’ll learn about…

…and why

Parametric equations can be used to obtain graphs of relations and functions.

Slide 1- 33

Relations

A relation is a set of ordered pairs (x, y) of real numbers.

The graph of a relation is the set of points in a

plane that correspond to the ordered pairs of the relation.

If x and y are functions of a third variable t, called a parameter, then the equations that define x and y are parametric equations.

Slide 1- 34

Parametric Curve, Parametric Equations

( ) ( )

( ) ( ) ( )( )

If and are given as functions

,

over an interval of -values, then the set of points , ,

defined by these equations is a . The equations are

of th

x y

x f t y g t

t x y f t g t

= =

=

parametric curve

parametric equations e curve.

Lines, line segments and many other curves can be defined parametrically.

General parametric equations involving angular measure:x = v0 cosθt and y = -16t2 + v0 sinθt + s

Ex. 1: Consider the path followed by an object that is propelled into the air as an angle of 45 degrees with an initial velocity of 48 ft/sec. The object will follow a parabolic path.

Write a Cartesian equation and a set of parametric equations to model this example.

Slide 1- 35

Graph each set of parametric equations, then find the Cartesian equation relating the variables (eliminate the parameter):

x = 2t + 1

y = 2 – t

Cartesian Equation:Slide 1- 36

t 0 1 2

x

y

Slide 1- 37

x = r2 – 3r + 1y = r + 1

Cartesian Equation:

r

x

y

x = sin ry = cos r

Cartesian Equation:

Slide 1- 38

r

x

y

x = t3

y = t2/2

Slide 1- 39

t

x

y