Relations and Functions (Algebra 2)

Relations and Functions

 Analyze and graph relations.
 Find functional values.

1) ordered pair 8) function
2) Cartesian Coordinate 9) mapping
3) plane 10) one-to-one function
4) quadrant 11) vertical line test
5) relation 12) independent variable
6) domain 13) dependent variable
7) range 14) functional notation


This table shows the average lifetime Average Maximum
and maximum lifetime for some animals. Animal Lifetime Lifetime
(years) (years)

Cat 12 28

Cow 15 30

Deer 8 20

Dog 12 20

Horse 20 50


(years) (years)
The data can also be represented as
ordered pairs. Cat 12 28

Cow 15 30

Deer 8 20

Dog 12 20

Horse 20 50


(years) (years)
The ordered pairs for the data are:
Cow 15 30

Deer 8 20

Dog 12 20

Horse 20 50


(years) (years)
Cow 15 30
(12, 28), (15, 30), (8, 20),

(12, 20), and (20, 50) Deer 8 20

Dog 12 20

Horse 20 50


(years) (years)
Cow 15 30
(12, 28), (15, 30), (8, 20),

(12, 20), and (20, 50) Deer 8 20

The first number in each ordered pair Dog 12 20
is the average lifetime, and the second
number is the maximum lifetime. Horse 20 50


(years) (years)
Cow 15 30
(12, 28), (15, 30), (8, 20),

(12, 20), and (20, 50) Deer 8 20


(20, 50)


(years) (years)
Cow 15 30
(12, 28), (15, 30), (8, 20),

(12, 20), and (20, 50) Deer 8 20


(20, 50)
average
lifetime


(years) (years)
Cow 15 30
(12, 28), (15, 30), (8, 20),

(12, 20), and (20, 50) Deer 8 20


(20, 50)
average maximum
lifetime lifetime


You can graph the ordered pairs below Animal Lifetimes
on a coordinate system with two axes.
y
60

50

Maximum Lifetime
40

30

20

10

0 x
0 5 10 15 20 25 30

Average Lifetime


y
(12, 28), 60

50

Maximum Lifetime
40

30

20

10

0 x
0 5 10 15 20 25 30

Average Lifetime


y
(12, 28), (15, 30), 60

50

Maximum Lifetime
40

30

20

10

0 x
0 5 10 15 20 25 30

Average Lifetime


y
(12, 28), (15, 30), (8, 20), 60

50

Maximum Lifetime
40

30

20

10

0 x
0 5 10 15 20 25 30

Average Lifetime


y
(12, 28), (15, 30), (8, 20), 60

(12, 20), 50

Maximum Lifetime
40

30

20

10

0 x
0 5 10 15 20 25 30

Average Lifetime


y
(12, 28), (15, 30), (8, 20), 60

(12, 20), and (20, 50) 50

Maximum Lifetime
40

30

20

10

0 x
0 5 10 15 20 25 30

Average Lifetime


y
(12, 28), (15, 30), (8, 20), 60

(12, 20), and (20, 50) 50

Maximum Lifetime
40
Remember, each point in the coordinate
plane can be named by exactly one 30
ordered pair and that every ordered pair
names exactly one point in the coordinate 20

plane.
10

0 x
0 5 10 15 20 25 30

Average Lifetime


y
(12, 28), (15, 30), (8, 20), 60

(12, 20), and (20, 50) 50

Maximum Lifetime
40
Remember, each point in the coordinate
plane can be named by exactly one 30
ordered pair and that every ordered pair
names exactly one point in the coordinate 20

plane.
10

The graph of this data (animal lifetimes) 0 x
lies in only one part of the Cartesian 0 5 10 15 20 25 30

coordinate plane – the part with all Average Lifetime
positive numbers.

The Cartesian coordinate system is composed of the x-axis (horizontal),

-5 0 5

and the y-axis (vertical), which meet at the origin (0, 0) and divide the plane into
four quadrants.

5

Origin
(0, 0)

0
-5 0 5

-5

four quadrants.
You can tell which quadrant a point is in by looking at the sign of each coordinate of
the point.

5

Quadrant II Origin
Quadrant I
( --, + ) ( +, (0, )0)
+

0
-5 0 5

Quadrant III Quadrant IV
( --, -- ) ( +, -- )

-5

four quadrants.
You can tell which quadrant a point is in by looking at the sign of each coordinate of
the point.

5

Quadrant II Origin
Quadrant I
( --, + ) ( +, (0, )0)
+

0
-5 0 5

Quadrant III Quadrant IV
( --, -- ) ( +, -- )

-5

The points on the two axes do not lie in any quadrant.

In general, any ordered pair in the coordinate plane can be written in the form (x, y)


A relation is a set of ordered pairs, such as the one for the longevity of animals.



The domain of a relation is the set of all first coordinates (x-coordinates) from the
ordered pairs.



ordered pairs.
The range of a relation is the set of all second coordinates (y-coordinates) from the
ordered pairs.



ordered pairs.
The range of a relation is the set of all second coordinates (y-coordinates) from the
ordered pairs.

The graph of a relation is the set of points in the coordinate plane corresponding to the
ordered pairs in the relation.

A function is a special type of relation in which each element of the domain is paired
with ___________ element in the range.

exactly one

exactly one

A mapping shows how each member of the domain is paired with each member in
the range.

exactly one

the range.

Functions
   3,1 ,  0,2 ,  2,4

exactly one

the range.

Functions
   3,1 ,  0,2 ,  2,4
Domain

-3

0

2

exactly one

the range.

Functions
   3,1 ,  0,2 ,  2,4
Domain Range

-3 1

0 2

2 4

exactly one

the range.

Functions
   3,1 ,  0,2 ,  2,4
Domain Range

-3 1

0 2

2 4

one-to-one function

exactly one

the range.

Functions
   1,5 , 1,3 ,  4,5

exactly one

the range.

Functions
   1,5 , 1,3 ,  4,5
Domain Range

-1
5
1
3
4

exactly one

the range.

Functions
   1,5 , 1,3 ,  4,5
Domain Range

-1
5
1
3
4

function,
not one-to-one

exactly one

the range.

Functions
  5,6 ,   3,0 , 1,1 ,   3,6

exactly one

the range.

Functions
  5,6 ,   3,0 , 1,1 ,   3,6
Domain Range

5 6

-3 0

1 1

exactly one

the range.

Functions
  5,6 ,   3,0 , 1,1 ,   3,6
Domain Range

5 6

-3 0

1 1

not a function

State the domain and range of the relation shown y

in the graph. Is the relation a function?
(-4,3) (2,3)

x

(-1,-2) (3,-3)

(0,-4)


(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
x

(-1,-2) (3,-3)

(0,-4)


(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is: x

(-1,-2) (3,-3)

(0,-4)


(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is: x

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

(0,-4)


(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is: x

{ -4, -1, 0, 2, 3 } (-1,-2) (3,-3)
The range is:
(0,-4)


(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is: x

{ -4, -1, 0, 2, 3 } (-1,-2) (3,-3)
The range is:
(0,-4)
{ -4, -3, -2, 3 }


(-4,3) (2,3)
The relation is:
{ (-4, 3), (-1, 2), (0, -4), (2, 3), (3, -3) }
The domain is: x

{ -4, -1, 0, 2, 3 } (-1,-2) (3,-3)
The range is:
(0,-4)
{ -4, -3, -2, 3 }

Each member of the domain is paired with exactly one member of the range,
so this relation is a function.

You can use the vertical line test to determine whether a relation is a function.


Vertical Line Test

If no vertical line intersects a
graph in more than one point,
the graph represents a function.

y

x


Vertical Line Test

If no vertical line intersects a If some vertical line intercepts a
graph in more than one point, graph in two or more points, the
the graph represents a function. graph does not represent a function.

y y

x x

The table shows the population of Indiana over the last several Population
Year
decades. (millions)

1950 3.9

1960 4.7

1970 5.2

1980 5.5

1990 5.5

2000 6.1

Year
decades. (millions)

1950 3.9

We can graph this data to determine 1960 4.7
if it represents a function.
1970 5.2

1980 5.5

1990 5.5

2000 6.1

Year
decades. (millions)

1950 3.9

1970 5.2
Population of Indiana
8 1980 5.5
7

6
1990 5.5
Population

5
(millions)

2000 6.1
4

3

2

1

0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year

Year
decades. (millions)

1950 3.9

1970 5.2
8 1980 5.5
7

6
1990 5.5
Use the vertical
Population

5
(millions)

line test. 2000 6.1
4

3

2

1

0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year

Year
decades. (millions)

1950 3.9

1970 5.2
8 1980 5.5
7

6
1990 5.5
Use the vertical
Population

5
(millions)

line test. 2000 6.1
4

3
Notice that no vertical line can be drawn that
2
contains more than one of the data points.
1

0
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year

Year
decades. (millions)

1950 3.9

1970 5.2
8 1980 5.5
7

6
1990 5.5
Use the vertical
Population

5
(millions)

line test. 2000 6.1
4

3
Notice that no vertical line can be drawn that
2
contains more than one of the data points.
1

0 Therefore, this relation is a function!
‘50 ‘60 ‘70 ‘80 ‘90 ‘00 7
0
Year

Graph the relation y  2 x  1

1) Make a table of values.


x y
-1 -1

0 1

1 3

2 5

Graph the relation y  2 x  1 2) Graph the ordered pairs.

x y
-1 -1

0 1

1 3

2 5

1) Make a table of values. y
7

6
x y 5

4
-1 -1
3

2
0 1
1

0 x
1 3
-1

2 5 -2

-3
-5 -4 -3 -2 -1 1 2 3 4 5
0

7

6
x y 5

4
-1 -1
3

2
0 1
1

0 x
1 3
-1

2 5 -2

-3
-5 -4 -3 -2 -1 1 2 3 4 5
0

3) Find the domain and range.

7

6
x y 5

4
-1 -1
3

2
0 1
1

0 x
1 3
-1

2 5 -2

-3
-5 -4 -3 -2 -1 1 2 3 4 5
0

Domain is all real numbers.

7

6
x y 5

4
-1 -1
3

2
0 1
1

0 x
1 3
-1

2 5 -2

-3
-5 -4 -3 -2 -1 1 2 3 4 5
0

Range is all real numbers.

7

6
x y 5

4
-1 -1
3

2
0 1
1

0 x
1 3
-1

2 5 -2

-3
-5 -4 -3 -2 -1 1 2 3 4 5
0

3) Find the domain and range. 4) Determine whether the relation is a function.

7

6
x y 5

4
-1 -1
3

2
0 1
1

0 x
1 3
-1

2 5 -2

-3
-5 -4 -3 -2 -1 1 2 3 4 5
0

Domain is all real numbers. The graph passes the vertical line test.

7

6
x y 5

4
-1 -1
3

2
0 1
1

0 x
1 3
-1

2 5 -2

-3
-5 -4 -3 -2 -1 1 2 3 4 5
0

Domain is all real numbers. The graph passes the vertical line test.
For every x value there is exactly one y value,
so the equation y = 2x + 1 represents a function.


Graph the relation x  y 2  2



x y
2 -2

-1 -1

-2 0

-1 1

2 2


Graph the relation x  y 2  2 2) Graph the ordered pairs.

x y
2 -2

-1 -1

-2 0

-1 1

2 2


y
1) Make a table of values. 7

6

5
x y
4

2 -2 3

2

-1 -1 1

0 x

-2 0 -1

-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2


y

6

5
x y
4

2 -2 3

2

-1 -1 1

0 x

-2 0 -1

-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2



y

6

5
x y
4

2 -2 3

2

-1 -1 1

0 x

-2 0 -1

-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2

Domain is all real numbers,
greater than or equal to -2.


y

6

5
x y
4

2 -2 3

2

-1 -1 1

0 x

-2 0 -1

-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2



y

6

5
x y
4

2 -2 3

2

-1 -1 1

0 x

-2 0 -1

-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
2 2 4) Determine whether the relation is a function.


y

6

5
x y
4

2 -2 3

2

-1 -1 1

0 x

-2 0 -1

-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range. The graph does not pass the vertical line test.


y

6

5
x y
4

2 -2 3

2

-1 -1 1

0 x

-2 0 -1

-2
-1 1 -3
-5 -4 -3 -2 -1 1 2 3 4 5
0
3) Find the domain and range. The graph does not pass the vertical line test.
Domain is all real numbers, For every x value (except x = -2),
greater than or equal to -2. there are TWO y values,
Range is all real numbers. so the equation x = y2 – 2
DOES NOT represent a function.


When an equation represents a function, the variable (usually x) whose values make
up the domain is called the independent variable.


The other variable (usually y) whose values make up the range is called the
dependent variable because its values depend on x.



Equations that represent functions are often written in function notation.




The equation y = 2x + 1 can be written as f(x) = 2x + 1.




y
The symbol f(x) replaces the __ , and is read “f of x”




y
The symbol f(x) replaces the __ , and is read “f of x”
The f is just the name of the function. It is NOT a variable that is multiplied by x.


Suppose you want to find the value in the range that corresponds to the element
4 in the domain of the function.
f(x) = 2x + 1


f(x) = 2x + 1

This is written as f(4) and is read “f of 4.”


f(x) = 2x + 1

The value f(4) is found by substituting 4 for each x in the equation.


f(x) = 2x + 1


Therefore, if f(x) = 2x + 1


f(x) = 2x + 1


Then f(4) = 2(4) + 1


f(x) = 2x + 1


Then f(4) = 2(4) + 1
f(4) = 8 + 1


f(x) = 2x + 1


Then f(4) = 2(4) + 1
f(4) = 8 + 1
f(4) = 9


f(x) = 2x + 1


Then f(4) = 2(4) + 1
f(4) = 8 + 1
f(4) = 9

NOTE: Letters other than f can be used to represent a function.


f(x) = 2x + 1


Then f(4) = 2(4) + 1
f(4) = 8 + 1
f(4) = 9

NOTE: Letters other than f can be used to represent a function.
EXAMPLE: g(x) = 2x + 1


Given: f(x) = x2 + 2 and g(x) = 0.5x2 – 5x + 3.5
Find each value.


Find each value.

f(-3)


Find each value.

f(-3)

f(x) = x2 + 2


Find each value.

f(-3)

f(x) = x2 + 2
f(-3) = (-3)2 + 2


Find each value.

f(-3)

f(x) = x2 + 2
f(-3) = (-3)2 + 2
f(-3) = 9 + 2


Find each value.

f(-3)

f(x) = x2 + 2
f(-3) = (-3)2 + 2
f(-3) = 9 + 2
f(-3) = 11


Find each value.

f(-3) g(2.8)
f(x) = x2 + 2
f(-3) = (-3)2 + 2
f(-3) = 9 + 2
f(-3) = 11


Find each value.

f(-3) g(2.8)
f(x) = x2 + 2 g(x) = 0.5x2 – 5x + 3.5
f(-3) = (-3)2 + 2
f(-3) = 9 + 2
f(-3) = 11


Find each value.

f(-3) g(2.8)
f(x) = x2 + 2 g(x) = 0.5x2 – 5x + 3.5
f(-3) = (-3)2 + 2 g(2.8) = 0.5(2.8)2 – 5(2.8) + 3.5
f(-3) = 9 + 2
f(-3) = 11


Find each value.

f(-3) g(2.8)
f(x) = x2 + 2 g(x) = 0.5x2 – 5x + 3.5
f(-3) = (-3)2 + 2 g(2.8) = 0.5(2.8)2 – 5(2.8) + 3.5
f(-3) = 9 + 2 g(2.8) = 3.92 – 14 + 3.5
f(-3) = 11


Find each value.

f(-3) g(2.8)
f(x) = x2 + 2 g(x) = 0.5x2 – 5x + 3.5
f(-3) = (-3)2 + 2 g(2.8) = 0.5(2.8)2 – 5(2.8) + 3.5
f(-3) = 9 + 2 g(2.8) = 3.92 – 14 + 3.5
f(-3) = 11 g(2.8) = – 6.58


Given: f(x) = x2 + 2
Find the value.


Find the value.

f(3z)


Find the value.

f(3z)

f(x) = x2 + 2


Find the value.

f(3z)

f(x) = x2 + 2
f( 3z ) = (3z) 2 + 2


Find the value.

f(3z)

f(x) = x2 + 2
f( 3z ) = (3z) 2 + 2
f(3z) = 9z2 + 2

Relations and Functions (Algebra 2)

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