8-22-13 7-1,7-2 review a personal approach math 94

8-22-13

7-1,7-2 review

A personal approach

Math 94

Warm Up Describe the domain of each. Use appropriate notation.

Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Find each function value for the function f given by

a) f(3) b) f(2) c) f(7)

Solution

a) FOR f(-3) use f(x) = x + 3: f(3) = 3 + 3 = 0

b) FOR f(2) use f(x) = x2; f(2) = 22 = 4

c) FOR f(7) use f(x) = 4x = 4(7) = 28

2

3, if 3

( ) , if 3 4

4 , if 4

x x

f x x x

x x

Example

Eaxmple

Think of three married couples you know. If you cannot think of any make them up. Write them down.

Chris and Melanie, Dugg and Diana, Brad and Kathi

Now write three ordered pairs describing the relationships.

(Chris, Melanie) (Dugg, Diana) (Brad, Kathi)

Relation

This is an example of a relation.


Another relation is

(1,2), (3,4), (5,6)

The entries are called “coordinates”. 1 is the first coordinate, 2 is the second. Chris is the first coordinate, Melanie is the second.

6Copyright © 2010 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Relation A relation is

-a set of ordered pairs.

-a correspondence between a first set called the domain, and a second set, called the range.

Note that for each member of the domain there is at least one member of the range.

Think-If I only have one person, is that a relationship?

Domain and Range

Notice the way my ordered pairs are written all the husbands are on the left and the wives are on the right.


The husbands are the domain and the wives are the range.

It is a relation because for EVERY husband there is a wife.

Domain and Range

In this example (1,2), (2,3), (3,4), (4,5) my domain is {1,2,3,4} and my range is {2,3,4,5}

Remember ordered pairs come in (x, y) form s the ones on left are x’s and the ones on the right are y’s.

This is why we can say the domain is “all the x’s” and the range is “all the y’s”

Other ways to write relations

Dugg

Correspondence =

Married To

Chris

BradKathi

Diana

Melanie

RangeDomain


Dugg

Chris

Brad

Kathi

Diana

Melaniee


Dugg

Chris

Kathi

Diana

Melanie

Brad

x y

Independent Variable, Dependent Variable

Now back to my relation


Think who really depends on who. The wife depends on the husband for security and being taken care of. So the second coordinate depends on the first.

Dependable Needy

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=p11Q-BabIXLBTM&tbnid=FxiEIkilCdPTTM:&ved=0CAUQjRw&url=http://rhill.coe.uga.edu/workethic/less5.htm&ei=mHUWUpHgKIGwjAKwvICQCg&psig=AFQjCNHtLd_YhgY7z6rL_5pR_LlU34VKEg&ust=1377289776736461

Which depends on which?

The idea of dependence is what functions are about.


Function A function is

-a dependence relation.

-A relation where y depends on x written y(x).

-Since it is a function we replace y with f and write f(x).

More on functions

A relation where for any member of the domain, there is exactly one member of the range.

-This is also stated as for every x there is only one y.

Marriage is a good example because for each husband there is only one wife.


Unless

You live in a place where polygamy is legal.

(Mike, Lisa) (Mike, Sally) (Mike, Sue)

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=usP9Dkm7ROVLkM&tbnid=1vvG3C5lA7g8-M:&ved=0CAUQjRw&url=http://puremormonism.blogspot.com/2010/06/why-im-abandoning-polygamy.html&ei=SIcWUsDwC4KhigKG6oGoBA&bvm=bv.51156542,d.cGE&psig=AFQjCNF2lz946DFAbfiIrsfux_0A9tdWnA&ust=1377294510590665

http://www.google.com/url?sa=i&rct=j&q=&esrc=s&frm=1&source=images&cd=&cad=rja&docid=usP9Dkm7ROVLkM&tbnid=1vvG3C5lA7g8-M:&ved=0CAUQjRw&url=http://puremormonism.blogspot.com/2010/06/why-im-abandoning-polygamy.html&ei=SIcWUsDwC4KhigKG6oGoBA&bvm=bv.51156542,d.cGE&psig=AFQjCNF2lz946DFAbfiIrsfux_0A9tdWnA&ust=1377294510590665

http://www.google.com/url?sa=i&rct=j&q=surprised+face&source=images&cd=&cad=rja&docid=s5i5xuzjV1qw5M&tbnid=EcVVSSpLogrNzM:&ved=0CAUQjRw&url=http://www.nwscholasticpress.org/2012/10/09/write-great-leads-that-will-grab-your-readers-attention-by-knowing-these-9-effective-strategies/&ei=zrEWUpzIGaqViQKftIDoDw&bvm=bv.51156542,d.cGE&psig=AFQjCNENcRBrUOUkG4cG7ajO_0MKBCkj6g&ust=1377305340179344


The following relation is presented in two forms, map form and table form. Determine if the correspondence is a function.

8 0–5

2

17

SolutionThe correspondence is a function because each member of the domain corresponds to exactly one member of the range.

Example

x y

8 2

0 17

-5 17


Determine if the correspondence is a function.Write the relation in table form.

JacksonMaxCade

Football

Wrestling

Soccer

SolutionThe correspondence is not a function because a member of the domain (Jackson) corresponds to more than one member of the range.

Example


Determine whether the correspondence is a function.

A set of rectangles

Range

Solution

The correspondence is a function, because each rectangle has only one area.

Each rectangle’s area

A set of numbers

Domain Correspondence

Example


Determine if the correspondence is a function.

Famous singers

Range

Solution

The correspondence is not a function, because some singers have recorded more than one song.

A song that the singer has recorded

A set of song titles

Domain Correspondence

Example


Graph formIf a function is a set of ordered pairs of numbers we can draw it in graph form.

A function is

in graph form

if the ordered

pairs are

plotted.

{(-5,1), (1,0), (4,3),(3,-5)}

6

54

2

3

-4

1

-2

-1

-3

-5 -4 -3 -2 -1 1 2 3 4 5

f

-5

Why is this a function?


Find the domain and range of the function f below.

6

54

2

3

-4

1

-2

-1

-3

-5 -4 -3 -2 -1 1 2 3 4 5

f

-5

Example


SolutionHere f can be written {(–5, 1), (1, 0), (3, –5), (4, 3)}. The domain is the set of all first coordinates, {–5, 1, 3, 4}, and the range is the set of all second coordinates, {1, 0, –5, 3}.

6

54

2

3

-4

1

-2

-1

-3

-5 -4 -3 -2 -1 1 2 3 4 5

f

-5


Find the coordinateIt is common to ask which member of one set corresponds to a member of another. Return to my function. (Chris, Melanie) (Dugg , Diana) (Brad , Kathi). Which wife corresponds to Chris? Why? Note that this is like saying which member of the range corresponds to Chris. Chris is an x so we find the y that goes with Chris.

Which member of the domain corresponds to Diana?

Why is this a function?

6

54

2

3

-4

1

-2

-1

-3

-5 -4 -3 -2 -1 1 2 3 4 5

f

-5

{(-5,1), (1,0), (4,3),(3,-5)}

Which member of the domain corresponds to 0? Which member of the range corresponds to -5?


For the continuous function f represented below, determine each of the following.

a) What member of the range is paired with -2

b) What member of the domain is paired with 4

c) An x value for which f(x) = 3

y

x -5 -4 -3 -2 -1 1 2 3 4 5

f4

-2

-1

-4

-3

32

5

1

6

7

Example


a) What member of the range is paired with -2

Solution

The question is saying what member of the range is paired with -2 which means what y value corresponds to x = -2. So you are looking for a y value. Find x = -2 on the horizontal axis and go to the graph. The y-coordinate of the point is 3. Therefore 3 is the member of the range paired with -2.

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

f4

-2

-1

-4

-3

32

5

1

6

Input

Output

7


b) What member of the domain is paired with 4

Solution

The question is saying what member of the domain is paired with 4 which means what x value corresponds to y = 4. So you are looking for a x value. Find y = 4 on the graph and go to the x axis. The x-coordinate of the point is 1. Therefore 1 is the member of the domain paired with 4.

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

f4

-2

-1

-4

-3

32

5

1

6

Input

Output7

How is this graph different from the previous example?


The Vertical-Line TestThis is a test to see if a graph is a function. It is more often used on continuous graphs.

If it is possible for a vertical line to cross a graph more than once, then the graph is not the graph of a function.

When a vertical line intersects more than once it represents multiple inputs with the same output.


Recall a graph is a set of ordered pairs. So graphs that do not represent functions are still relations.

A function. Every vertical line intersects at most once.

Not a function. Two y-values correspond to one x-value

Not a function. Three y-values correspond to one x-value


Functions and DependenceOne example of a function is a soda machine. The sodas (outputs) depend on the money (inputs).

Can you think of another example of a function?


A nice visual

The function pictured has been named f. Here x is an input, and f (x) – read “f of x,” is the corresponding output.

With this notation y = f (x).


Function Notation and EquationsIn math most functions are described by equations.

For example, f (x) = 5x +2 describes the function

that takes an input x, multiplies it by 5 and then

adds 2. f (x) = 5x + 2

To calculate the output f (3), take the input 3, multiply it by 5, and add 2 to get 17. That is, substitute 3 into the formula for f (x).

Input

f (3) = 5(3) + 2 = 17 Output

When I study my learning depends on my

effort. Thus learning is a function of

effort or L = f(e).

y depends on x so y = f(x).


Solution

Find each indicated function value.

a) f (–2), for f (x) = 3x 2 + 2x

b) g(4), for g(t) = 6t + 9

c) h(m +2), for h(x) = 8x + 1

a) f (–2) = 3(–2)2 + 2(–2) = 12 – 4 = 8

b) g(4) = 6(4) + 9 = 24 + 9 = 33

c) h(m +2) = 8(m+ 2) + 1 = 8m + 16 + 1

= 8m + 17

Example


Note that whether we write f (x) = 5x +2 or f (m) = 5m +2, we still have f (3) = 17. Thus the independent variable can be thought of as a dummy variable.

When a function is described by an equation, the domain is often unspecified. In such cases, the domain is the set of all numbers for which function values can be calculated.


A small business started out in the year 1996 with 10 employees. By the start of 2000 there were 28 employees, and by the beginning of 2004 the business had grown to 34 employees. Estimate the number of employees in 1998 and also predict the number of employees in 2007.

Example


SolutionWrite the relation in ordered pair form and graph form. Use the graph to answer the question.

Plot the points and connect the three points. Let the horizontal axis represent the year and the vertical axis the number of employees. Label the function itself E.

10

3020

4050

2004

2000

1996

Num

ber

of

Em

ploy

ees

Year


3. Using the graph.

To estimate the number of employees in 1998, locate the point directly above the year 1998. Then estimate its second coordinate by moving horizontally from that point to the y-axis. We see that

10

3020

4050

2004

2000

1996

19

1998

Year

Num

ber

of

Em

ploy

ees

(1998) 19.E


3. Using the graph (continued).

To predict the number of employees in 2007, extend the graph and extrapolate. We see that

10

3020

4050

2004

2000

1996

2007

Year

Num

ber

of

Em

ploy

ees

(2007) 40.E


4. Check.

A precise check would involve knowing more information. Since 19 is between 10 and 28 and 40 is greater than 34, the estimate seems plausible.

5. State.

In 1997, there were about 19 employees at the small business. By 2007, the number of employees should grow to 40.


When a function is in ordered pair form, the domain is the set of all first coordinates and the range is the set of all second coordinates.

Find the domain and range for the function f given by

f = {(–5, 1), (1, 0), (3, –5), (4, 3)}.

Solution

The domain is the set of all first coordinates:{–5, 1, 3, 4}.

The range is the set of all second coordinates:{1, 0, –5, 3}.

Example


Find the domain and range of the function f in continuous graph from. y

x -5 -4 -3 -2 -1 1 2 3 4 5

f4

-2

-1

-4

-3

32

5

1

6

7

Example


Solution

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

f4

-2

-1

-4

-3

32

5

1

6

The domain of f

7

The domain of f

The domain of f is the set of all x-values of the points on the curve. These extend continuously from -5 to 3 and can be viewed as the curve’s shadow, or projection, on the x-axis. Thus the domain in set interval notation is

{ | 5 3}.x x

the domain is the set of all first coordinates and the range is the set of all second coordinates.


The range of f

Solution

x

y

-5 -4 -3 -2 -1 1 2 3 4 5

f4

-2

-1

-4

-3

32

5

1

6

The range of f

7

{ | 1 7}.y y

The range of f is the set of all y-values of the points on the curve. These extend continuously from -1 to 7 and can be viewed as the curve’s shadow, or projection, on the y-axis. Thus the range in set interval notation is

the domain is the set of all first coordinates and the range is the set of all second coordinates.


Recall

When a function is described by an equation, the domain is often unspecified. In such cases, the domain is the set of all numbers for which function values can be calculated.


Determine the domain of 2( ) 3 4.f x x

Solution

We ask, “Is there any number x for which we cannot compute 3x2 – 4?” Since the answer is no, the domain of f is the set of all real numbers.

,

Example


Determine the domain of Solution

We ask, “Is there any number x for which

cannot be computed?” Since cannot 2

8x be computed when x – 8 = 0 the answer is yes.

x – 8 = 0,x = 8

Thus 8 is not in the domain of f, whereas all other real numbers are. The domain of f is

{ | is a real number 8}.x x and x

28x

2( ) .

8f x

x

To determine what x-value would cause x – 8 to be 0, we solve an equation:

Example


Functions Defined PiecewiseA piecewise function is a function whose

equation differs according its domain. These functions are piecewise defined.

To find f(x) for a piecewise function

a) Determine what part of the domain x belongs to.

b) Then use the equation for that part of the domain.

2

3, if 3

( ) , if 3 4

4 , if 4

x x

f x x x

x x

Quiz

Quiz

8-22-13 7-1,7-2 review a personal approach math 94

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