Functione and Relations - Krahnvmc



Unit: Relations and Functions

Name ________________________ Dates Taught _________________

Relations and Functions

|General Outcome | | | |

|10I.R.1: |Interpret and explain the relationship among data, graphs and context. | | |

| |Match corresponding representations of data, graphs and contexts. | | |

| |Graph, with or without technology, a set of data, and determine the restrictions on the | | |

| |domain and range. | | |

| |Describe the restrictions on the domain and range for a context. | | |

|10I.R.2: |Demonstrate an understanding of relations and functions. | | |

|10I.R.2: |Determine, and express in a variety of ways, the domain & range of a relation. | | |

Comments : ________________________________________________

_____________________________________________________________

_____________________________________________________________

_____________________________________________________________

_____________________________________________________________

Outcome 10I.R.1: Data and Graphs

Example Jenna is a cross-country skier. On Saturday, she skied the Blue Ptarmigan Trail in one hour. The graph shows her speed as she went along the trail. She started off along the trail at 10:00 am.

What is her speed at 10:10 am? __________

At what time is she going fastest and what is her speed at that time?

time _________ speed ________

What happens to her speed between 10:25 am and 10:35 am?

What is her speed at 10:40 am? __________

Describe what the trail is like and what her trip was like from the evidence of the graph. Note that the speed depends on the time or that the speed is a function of time.

Time is the independent variable and speed is the dependent variable.

From the graph complete these statements:

a) At point A, t = ________ and speed = ________

b) Coordinates of point A are ( ______, ______). B are ( ______, ______).

C are ( ______, ______).

Example A company cafeteria has a large vending machine that sells cans of soft drinks. The graph below shows the number of cans in the machine on a typical day.

a) Describe how the number of cans in the machine varies during the day.

b) When are morning coffee breaks and lunch-times?

c) What happens just before lunch-time?

d) Can employees use the drink machine during working hours?

e) How many cans of drinks were sold during this day?

f) What appears to be the working hours for this company? How do you know?

Outcome 10I.R.1: Data and Graphs (2)

• The shape of a graph can tell us a great deal even if there are no numbers on the axes. For example, the set of graphs below show how interest rates have changed over one year in three different countries.

The first graph shows that interest rates in that country rose steadily throughout the year.

The second graph shows that interest rates were high at the beginning of the year, remained constant for most of the year, and dropped rapidly for the last part of the year.

The third graph shows that interest rates increased rapidly at first, then more slowly to reach a maximum about halfway through the year, and decreased for the rest of the year.

Hints for interpreting graphs:

• When the value of the independent variable is zero, what is the value of the dependent variable? Does it make sense?

• As the value of the independent variable increases, does the value of the dependent variable increase or decrease? What does this mean for these particular variables? Does it make sense?

• Does the value of the dependent variable change at a steady rate? If not, how does it change? Is the change faster at first and slower later on . . . or is it slow at first and faster later on?

Example: Every morning at camp, one of the scouts hoists a flag to the top of a flagpole. The graphs below show the height of the flag as a function of time. Which do you think models the situation most realistically? If you think none of the graphs is realistic, draw your own version and explain it. Do any of the graphs represent an impossible situation.

Example: A group of city workers have to plant a large number of seedlings, Which of the following graphs models most realistically the relationship between the number of workers involved and the time it takes to complete the job. Explain your answer.

Homework: Handout “Functions: Interpreting and Creating Graphs”

Outcome 10I.R.1: Data and Graphs (3)

Example

• The graph shows a journey by car from Swan River to Dauphin and return.

a) How far is it from Swan River to Cowan?

b) How far is it from Cowan to Dauphin?

c) At which two places does the car stop?

d) How long does the car stop at Dauphin?

e) At what time does the car

i) arrive in Cowan?

ii) arrive back in Swan River?

f) At what speed is the car travelling

i) from Swan River to Cowan?

ii) from Cowan to Dauphin?

iii) from Dauphin to Swan River?

Example The graph shows trips by a bus and a car along the same road. The bus goes from Winnipeg to Portage and back to Winnipeg. The car goes from Portage to Winnipeg and back to Portage.

a) At what time did the bus and the car meet for the second time?

b) At what speed did the car travel from Portage to Winnipeg?

c) What was the average speed of the bus over its entire journey?

d) Approximately how far apart were the bus and the car at 9:45?

e) What was the greatest speed attained by the car during the entire journey?

Example The graph shows how the share price of the chemical firm ICI varied over a period of weeks. The share price is the number of cents paid for one share in the company.

0 4 8 12 16 20

a) What was the share price in week 4?

b) Naomi bough 200 shares in week 6 and sold them all in week 18. How much profit did she make?

c) Mr. Gibson can buy (and then sell) 500 shares. He consults a very accurate fortune teller who can predict the share price over coming weeks. What is the maximum profit he could make?

d) When there is a full moon the fortune teller’s predictions can be fairly disastrous. What is the maximum loss Mr. Gibson could make?

Homework:

Unit 6.1 (Graphs of Relations) Page 115 #1, 2, 7, 8, 9 (Answers P.267)

Pearson Unit 5.3 (Interpreting and Sketching Graphs)

Outcome 10I.R.2: Recognition of Functions vs. Relations

• A is any set of . It represents a relationship between or a correspondence between two variables.

• Relations may be represented in many ways. The most common being ordered pairs, a table of values, a graph or a mapping diagram.

Example: The relation written as a list { (2, 1), (3, 5), (6, -1), (7, 2)} could be represented as

i) a graph ii) a mapping diagram

iii) Table of Values

• A is a type of relation for which value in the domain corresponds with value.

Ex: { (2, 1), (3, 5), (6, -1), (7, 2)} is a because there is only one y for each x.

Ex: {(2, 3), (3, 1), (5, 4), (2, 6)} is a function because when x =2, there are 2 y-values.

Examples: Are these functions?

1) {(3, 8), (4, 9), (5, 10), (6, 11)} 2) {(-1, 7), (-1, 5), (-1, 3), (-1, 1)}

3) {(5, 9)} 4) {(3, 3), (4, 4), (4, 5), (5, 5), (5, 6)}

• To determine whether a graph is a function or not, we may apply the test. Since a function has one y-value for x-value, a line will cross the graph of a function in no more than point.

• If the vertical line crosses the graph of a relation in than point, the relation is a function.

Function Not a Function

Y y

x

x

5) 6)

A1 ? A2 ? A3 ? A4 ? A5 ? A6 ?

7) 8)

2 10

4 12

14

9) E = {2, 4, 5, 8}

Homework: Unit 6.4 (Functions)

Outcome10I.R.2: Domain & Range of Relations

• The is defined as the set of (usually represented by the letter ) or the component of the ordered pair.

• The is defined as the set of (usually represented by the letter ) or the component of the ordered pair.

• There are many different ways of indicating domain and range:

Set Notation Interval Notation

a) (x is part of the set of real numbers)

b) (x is greater than or equal to 6)

[pic]

[pic]

[pic]

c) (greater than -2 or less than 1)

(between -2 and 1)

Rules for what brackets to use:

• When simply listing numbers, use brackets.

Ex: State the domain and range for the following relation: {(1,2), (-2, 3), (3, 4)}

D= R=

• brackets includes the number.

Ex:

• brackets does not include the number.

Ex:

Examples: State the domain and range of the following relation. Also, list the ordered pairs.

1) D = R =

1 5

6 Ordered pairs:

3 7

2) State the domain (D) and range (R) of the following ordered pairs:

a) A1 = [pic] D = R =

b) A2 = [pic] D =

R =

c) A3 = [pic] D =

R =

3) State the domain and range of the following relations in interval notation:

D1 = D2 =

R1 = R2 =

D3 = D4 =

R3 = R4 =

D5 = D6 =

R5 = R6 =

Homework: Unit 6.3 (Domain & Range)

Outcome 10I.R.8: Function notation to represent a linear function.

• Various notations are used to denote functions. Some letters such as f, g and h are commonly used to represent functions. The following notations may be used:

f:x ( 2x + 2 f(x) = 2x + 2 y = 2x + 2

• For the following functions:

o Draw the graph;

o State the domain and range;

o Evaluate f(0); and

o Find the zeros of the function, if any.

( f(x) = x + 1

( [pic]

( f:x ( 3x

Homework: Exercise 54 (MCAL20S)

Outcome 10I.R.10: Restrictions on the domain and range

Example Use your graphing calculator to graph the following functions to find the Domain and Ranges of each:

a) [pic]

• Touch the following buttons (keys) on your calculator.

y = then X,T,Θ,n x2 + 4 Your graph should look like :

the screen should show Y1 = X2 + 4

Then touch ZOOM and 6

Domain

Range

b) [pic]

• Delete the previous function by touching the CLEAR button. Then touch the following buttons (keys) on your calculator.

y = then 1 / ( X,T,Θ,n – 3 )

• Change the window to produce an easier to read graph.

Touch WINDOW and enter the following information:

Xmin = -5 Xmax = 5 Xscl = 1

Ymin = -5 Ymax = 5 Yscl = 1

Then GRAPH

Domain

(note: 3 is a non-permissable value for x

and is not in the Domain of g(x) )

Range

(note: 0 is a non-permissable value for g(x) and is not in the Range of g(x) )

c) [pic]

Delete the previous function by touching the CLEAR button.

Then touch the following buttons (keys) on your calculator.

y = then 2nd x2 X,T,Θ,n – 2 )

Use the same window you use for the previous example.

Touch GRAPH

Domain

Range

Homework: (for each of the following start with the home screen – Zoom 6 )

1. Graph [pic] [pic]

Touch the following: 2nd TRACE ENTER 1

This “activates” the Value function. Then 3.2 and ENTER.

At the bottom of your screen it should show.

x = 3.2 y = 1.7888544

In function notation: f(3.2) = 1.788544 (to 6 decimal places) or

since [pic] [pic] = 1.788544

Find f(8.5) and f(-2) to three decimal places.

What value did you get for f(8.5) ?_________ What value did you get for f(-2) ? ___________

Did you use the correct negative sign? Make sure you used (– ) instead of – .

Was there still a problem?

Answers: f(8.5) = 2.915 f(-2) = (a blank spot next to y = )

f(-2) does not exist because -2 is not in the Domain of f(x). (You can’t take the square root of a negative number.)

Clear your [pic] in y = before beginning the next problem

2. On the same axes graph [pic] and [pic] and determine the domain and range for each.

[pic] Domain ________________ Range _________________

[pic] Domain ________________ Range _________________

Clear your y = before beginning the next problem

3. Determine by graphing which values of x are excluded from the domains of the following functions.

[pic] and [pic] (don’t forget to put brackets around the denominators)

Excluded value(s) for the Domain of g(x) ____________

Excluded value(s) for the Domain of h(x) ____________

4. Without using your calculator predict the excluded value(s) from the Domain of the following functions:

a) [pic] b) [pic]

c) [pic] d) [pic]

Answers:

2. [pic] Domain [ 5 , ∞ ) Range [ 0 , ∞ )

[pic] Domain (–∞ , 5 ] Range [ 0 , ∞ )

3. [pic] Excluded value(s) for the Domain of g(x) 4 (can’t divide by zero)

[pic] Excluded value(s) for the Domain of h(x) –2 (can’t divide by zero)

4. a) [pic] Excluded value(s) for the Domain of f(x) 5

b) [pic] Excluded value(s) for the Domain of g(x) –1 and 4

c) [pic] Excluded value(s) for the Domain of h(x)

any values of x < 3

d) [pic] Excluded value(s) for the Domain of j(x)

any values of x > 4

Note: The Domains for functions f, g, h, and j would be written:

Domain f [pic] (This is read: The real numbers x not equal to 5)

Domain g [pic] (This is read: The real numbers x not equal to -1 and 4)

Domain h [3, ∞)

Domain j (–∞, 4 ]

Problems

1. Investigate the domains and ranges of these functions:

a) [pic]

b) [pic]

c) [pic]

d) [pic]

2. Consider the relation [pic]. What would its graph look like?

In order to graph this relation solve for [pic].

[pic] then solve for y

[pic] and [pic] (recall if x2 = 9, x can be either +3 or -3)

Now graph both solutions for y and state the Domain and Range of the relation.

Why is this NOT a function?

Test Review Name ______________________ Date _______

1. (9 pts) Represent the relation defined by the rule y = 2x – 5 by a table of values (list), a mapping, and a graph. State the domain and range for the relation and tell whether the relation is a function or not.

Table of Values Mapping

0 2 -2 4 -4

Graph

Domain ___________

Range ____________

Function or not a Function

______________

2. (12 pts) Indicate which of the following are Function and which are NOT functions:

____________________ ____________________

____________________ ____________________

A = { (4, 2), (5, 5), (6, 7), (-4, 11)} B = { (1, 2), (2, 5), (3, 5), (1, 11)}

____________________ ____________________

C = { (1, 2), (2, 7), (3, 7), (4, 11)} D = { (1, 3), (2, 3), (3, 3), (4, 3)}

____________________ ____________________

______________ ______________ ______________

______________ ______________ ______________

4. (4 pts) What value(s) are excluded from the domain of the following functions:

a) [pic] b) [pic]

_____________ _____________

c) [pic] d) [pic]

_____________ _____________

5) (9 pts) Make a sketch of the following relations and state

the Domain and Range for each:

a) [pic]

Domain ______________ Range ____________

b) [pic]

Domain ______________ Range ____________

c) [pic]

Domain ______________ Range ____________

[pic]

[pic]

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