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click on the topic to go to that sectionTable of Contents
• Relationships and Functions
• Domain and Range
• Vertical Line Test
• Linear Vs. Non-Linear Functions
• Glossary
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State CapitalNew Jersey TrentonNew York Albany
Colorado DenverRhode Island Providence
x y5 72 9
0 8-4 6
Person GrandparentEric ArthurEric Ellie
Maria CamineMaria Susan
x y4 32 8
0 84 6
A function is a relation where every input (x) has exactly one output (y).
Let's look at a few examples of relations. Some are functions and some are not.
A relation is a set of ordered pairs.
Relationships and Functions
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State CapitalNew Jersey TrentonNew York Albany
Colorado DenverRhode Island Providence
Person GrandparentEric ArthurEric Ellie
Maria CamineMaria Susan
Relationships and FunctionsLet's look at these a little closer.
This is a function. Each state (input) has only one capital (output).
This is not a function. Each person (input) has more than one grandparent. (output).
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x y5 72 9
0 8-4 6
x y4 32 8
0 84 6
Relationships and Functions
Each number in the x (input) column has only one y (output) as an answer so it is a function.
Notice that 4 has an output of both 3 and 6. Therefore it cannot be a function.
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x y8 35 8
3 8-1 6
x y4 32 8
0 84 6
A B
Let's compare the last problem with another one. How do they differ? Are either of them functions?
Relationships and Functions
In table A, there is one x (4) that has two different y's (3 & 6). This is NOT a function.
In table B, there are two different x's (3 & 5) that have the same y (8). This is a function.
click to reveal
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SarahJohnDwayneAlanJada
June 8Oct. 28March 18Oct. 28Dec. 29
SarahJohnDwayneAlanJada
April 9Jan. 23Sept. 14Feb. 23May 5July 19
Consider this: Two people in a class (input) may have the same birthday (output). However, any one person (input) does not have more than one birthday (output).
Relationships and Functions
John and Alan have thesame birthday. That can happen. It is a function.
Dwayne's birthday is February 23 and September 14. This is not possible. It is not a function.
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Input Output
ABCD
1234
Relationships and Functions
Notice that there are different ways to express functions.
They may be in words, a table, ordered pairs, a graph, or like the last slide, a function map.
In a function map one side is the input and the other is output. An arrow extends from each input element to its matching output element.Example:
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x y2 63 74 85 9
Relationships and Functions
Functions can also be written as sets of ordered pairs. All of the x coordinates are the input and all of the y coordinates are the output.
To figure out if sets of ordered pairs are functions, make a function map or table first.
{(2, 6), (3, 7), (4, 8), (5, 9)}
This is a function because every input has exactly one output.
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1 Is the following a function?
YesNo
1234
5432
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2 Is the following a function?
YesNo
x y1 25 79 213 617 5
Ans
wer
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3 Is the following a function?
YesNo
A person's social security number
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4 Is the following a function?
YesNo
x y3 24 54 79 69 12
Ans
wer
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5 Is the following a function?
YesNo
Any number times 3.
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6 Is the following a function?
YesNo 1
234
567891011
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7 Is the following a function?
YesNo
{(1, 2), (2, 3), (3, 4), (4, 5)}
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8 Is the following a function?
YesNo
{(1, 2), (1, 3), (2, 4), (3, 4), (3, 5)}
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9 Is the following a function?
YesNo
City State
Englewood New Jersey
Englewood Colorado
Springfield Maryland
Springfield Illinois
Springfield Montana
Ans
wer
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10 A relationship between x and y is defined by the
equation , where x is the input and y is
the output. Which statements about the relationship
are true? Select each correct statement.
A y is a function of x.
B The graph of the relationship is a line.
C When the input is -3, the output is 4.
D When the input is -2, the output is 3.
E The y-intercept of the relationship is (0,1).From PARCC EOY sample test non-calculator #4
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11 When the input to a function is -2, the output is 4. Which statement about this function must be true?
A An input of -2 has infinitely many possible outputs
B An input of -2 has exactly one possible output
C An output of 4 has infinitely number of inputs
D A output of 4 has exactly one input
From PARCC EOY sample test non-calculator #10
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Days 1 2 3 4
$ Earned 125 42 98 106
Domain and Range
Any set of numbers that is the input of a relation or function is called the domain.
Any set of numbers that is the output of a relation or function is called the range.
Domain: {1, 2, 3, 4} Range: {42, 98, 106, 125}
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x 1 2 3 4y 6 7 8 9 1
2
34
6
78
9Domain Range
Domain and Range
Which numbers are part of the domain and which are part of the range?
Rule: x + 5 = y
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Domain Range1
83
10
5 127 14
9
16
Domain and Range
How about when we have ordered pairs?
Which numbers are in the domain? Which are in the range?
{(1,8), (3, 10), (5, 12), (7, 14), (9, 16)}
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12 Which numbers are in the domain?
246810
1020304050
A 2B 10C 12D 8E 20
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13 Which numbers are in the range?
A 2B 10C 12D 8E 20
246810
1020304050
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14 Which numbers are part of the domain?
A 12B 20C 24D 11E 9
{(24, 12), (22, 11), (20, 10), (18, 9)}
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15 Which numbers are part of the range?
A 12B 20C 24D 11E 9
{(24, 12), (22, 11), (20, 10), (18, 9)}
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Vertical Line Test
A function can also be expressed as a graph.
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Function Function Not a Function Not a Function
Vertical Line Test
In order to verify that a graph is a function you can use the Vertical Line Test.In order to be a function, there may be only one point that goes through any given vertical line on the graph.
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Slope = 0 Undefined slope
Vertical Line Test
A linear equation with a slope of zero is a function.
Prove that a linear equation with an undefined slope cannot be a function.
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16 Is this a function?
YesNo
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17 Is this a function?
YesNo
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18 Is this a function?
YesNo
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19 Is this a function?
YesNo
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20 Is this a function?
YesNo
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21 Is this a function?
YesNo
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22 Is this a function?
YesNo
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A Linear Function is a graph that is represented by a straight line. As an equation it is written in slope intercept form: y=mx+b (see Graphing Equations Unit).
Linear functions will never have an exponent larger than 1 in the equation.
y = 2x - 3 y = -1/4x +4
Linear Functions
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y = 2x +3
+1+1+1+1
+3+3+3+3
x y-2 -1-1 10 31 52 7
Linear Functions
Linear Functions are incremental.
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y = x 2 y = x 3 + 2x 2 - x
Non-Linear Function
A Non-linear Function is any function that is not represented as a straight line.
Non-linear functions often have exponents in their equation.
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y = x 2 + 2
+1+1+1+1
-3-1+1+3
x y-2 6-1 30 21 32 6
Non-Linear Function
Non-linear functions are not incremental.
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23 The following is a linear function.
TrueFalse
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24 The following is a linear function.
TrueFalse
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25 The following is a linear function.
TrueFalse
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26 The following is a linear function.y = 5x + 7
TrueFalse
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27 The following is a linear function.
TrueFalse
3 94 165 256 36
Ans
wer
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28 The following is a linear function
TrueFalse
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29 The following is a linear function.
TrueFalse
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30 The following is a function.
TrueFalse
y = 2x 2 + 3
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31 The following is a function.
TrueFalse
x 3 5 7 9y 13 17 21 25 A
nsw
er
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32 The graph of a nonlinear function is shown on the coordinate plane.
In the graph y is the function of x. When the input of the function is -4, what is the output of the function?
From PARCC EOY sample test non-calculator #14
A -5
B -1
C 1
D 5
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33 The graph shows y as a function of x.
Students type their answers here
From PARCC EOY sample test non-calculator #6
For each interval in the table, indicate whether the function is increasing, decreasing, or neither increasing nor decreasing over the interval.
Interval Increasing Descreasing Neither
-7<x<-3-3<x<-1-1<x<11<x<33<x<55<x<7
Ans
wer
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34 Classify each equation as defining y as a linear or non-linear function of x. Select one cell per column. Students type their answers here
From PARCC EOY sample test non-calculator #17
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Back to
Instruction
Domain
Any set of numbers that is the input of a relation or function.
Domain: {1,2,3,4}
Domain: {10,25,30,45}
Domain: {1,2,3,4}
# of people 10 25 30 45
$ Earned 100 420 980 1060
{(1,2.3),(2,5),(3,10),(4,11)}
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Instruction
Function
A relation where every input (x) has exactly one output (y).
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Instruction
Function MapA map where one side is the input and the other is output and an arrow extends from each input element to its matching output
element.
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Instruction
Linear Function
A graph that is represented by a straight line.
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Instruction
Non-Linear Function
Any function that is not represented as a straight line.
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Instruction
Range
Any set of numbers that is the output of a relation or function.
Range: {100,420,980,1060}
# of people 10 25 30 45
$ Earned 100 420 980 1060 {(1,2.3),(2,5),(3,10),(4,11)}
Range: {5,4,3,2}
Range: {2.3,5,10,11}
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Back to
Instruction
Vertical Line TestA test that shows if an equation is a function by
examining the graph of the function and seeing if there is only one point that goes through any
given vertical line on the graph.
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