FunctionsFunctionsBy Adrienne Calomino

Objectives:

to identify the domain and range of a relation - review;

to recognize from a table of values or from coordinate points if a relation is a function;

to recognize from the graph of a familiar relations whether it is a function;

to use the vertical line test to determine if a graph is that of a function;

to recognize algebraically if a relation is a function.

x or input value y or output value

Review domain and range of a relationWarm–up exercises

When is a relation a function?When each element in the domain maps to one

and only one element in the range.Does each element of the domain map to only

one element of the range below?

DogCatDuckLionPigRabbit

11

10

7

Domain Range

Is this a Function? Why or why not?

DogCatDuckLionPigRabbit

11

10

7

Domain Range

How about this?DogCatDuckLionPigRabbit

11 910 6 7 5

Domain Range

This is a particular type of function we will see again later.

Function

A function maps each element of the domain to one and only one element of the range

Which is a function? Why?

Domain Range

input output

1 2

1 3

2 3

3 5

4 5

4 6

5 7

Domain Range

input output

1 2

2 3

3 3

4 5

5 5

6 7

7 8

Table A Table BTable A

Which is a function? Why?

Cell phone use by country

Million Users

U.S. 219

China 438

U.K. 61

Mexico 47

Germany 72

Italy 72

France 48

Pakistan 48

Japan 95

Russian 120

Table A Table BTable A

Million Users Cell phone use by country

219 U.S.

438 China

61 U.K.

47 Mexico

72 Germany

72 Italy

48 France

48 Pakistan

95 Japan

120 Russian

Try coordinate points or ordered pairs

{ (0,1) (0,-1), (1,2), (2,1), (3,1), (4,1) }

{ (0,1) (1,-1), (2,2), (-2,1), (3,2), (4,1) }

Which is a function? Why?

Set A

Set B

State the domain and range.

1) Now try some from your worksheet

2) Group activity

• Make up two problems.

• Each can be a table or set of coordinate points.

• One should be a relation but not a function.

• The other should be a function.

• Do not include the answers.

• Give the problems to the others in your group and see if they get it correct.

• If not, explain the correct answer to them.

Definition We say “function”. We write f.We say “f of x”. We write f(x).

Where x is a value in the domain and f(x) is a value in the range.

f is the name of the function.   x, which is placed within the parentheses, is called the argument of the function.  It is upon the argument,x, that the function called f will "operate."

Mapping a function

Domain f(x) = 2x Range

input or x output or f(x)

1 f(1) 2

2 f(2) 4

3 f(3) 6

-1 f(-1) -2

-2 f(-2) -4

-3 f(-3) -6

0 f(0) 0

Domain f(x) = x2 Range

input or x output or f(x)

1 f(1) 1

-1 f(-1) 1

2 f(2) 4

-2 f(-2) 4

4 f(4) 16

-4 f(-4) 16

0 f(0) 0

Table A Table BWhy are these functions? Give an example of an argument of a function.What is the domain and range?

Now let’s relate coordinate points to a graph.Is this a function? (Remember every x must map

to one and only one y!)

Look at these graphs. Is it a function?How can you tell if each x value maps to only one

y value?

Vertical line test A method to test if there exists one and only one y value

for every x value.

You draw vertical lines.

If the vertical line crosses the graph only once, it passes the test.

Why does this work?

Is the graph a function? Why or why not?

BA

Notice that it is OK if two x-values map to the same y-value.

However the test has to work for every value of x.

Beware of the behavior of the graph in the extremes. It might look like it is becoming vertical, but it may just be increasing or decreasing very gradually.

How to determine if a relation is a function algebraically

Substitute y for f(x) Solve for y (if needed) Is there only one y value in the range for each x value in the domain?

Try f(x)= 2x + 3 y = 2x + 3

What type of relation is this?Is it a function?

How to determine if a relation is a function algebraically

Substitute y for f(x) Solve for y (if needed) Is there only one y value in the range for each x value in the domain?

Try f(x)= 2x2

y = 2x2

What type of relation is this?Is it a function?

Look at these familiar functions f(x)=x2 and f(x)=x3

y=x2 y=x3

How about x = y2 ? Solve for y y = ±√x Is this a function? Why or why not?

Note that we need to pay attention to the domain of the function f…..

Consider f(x) = 2x+1/x-1y = 2x+1/ x-1

◦ How is the domain is restricted. Why? { x| x ≠ 1 }

f(x) = x+ 1/ x-2

◦ How is the domain is restricted. Why? { x| x ≠ 2 }