To determine if a relation is a function

Objectives
1. To determine if a relation is a function.
2. To find the value of a function.

Functions
A function is a relation in which each element of
the domain is paired with exactly one element of
the range. Another way of saying it is that there is
one and only one output (y) with each input (x).
f(x)x y

Function Notation
Output
Input
Name of
Function
y = f x( )

Determine whether each relation is a function.
 1. {(2, 3), (3, 0), (5, 2), (4, 3)}
 YES, every domain is different!
f(x)
2 3
f(x)
3 0
f(x)
5 2
f(x)
4 3

Input Output
-3 3
1 -2
4 1
4
Identify the Domain and Range. Then tell if the
relation is a function.
Domain = {-3, 1,4}
Range = {3,-2,1,4}
Function?
No: input 1 is mapped onto
Both -2 & 1
Notice the set notation!!!

Identify the Domain and Range. Then tell if
the relation is a function.
Input Output
-3 3
1 1
3 -2
4
Domain = {-3, 1,3,4}
Range = {3,1,-2}
Function?
Yes: each input is mapped
onto exactly one output

A Relation can be represented by a set of ordered
pairs of the form (x,y)
Quadrant I
X>0, y>0
Quadrant II
X<0, y>0
Quadrant III
X<0, y<0
Quadrant IV
X>0, y<0
Origin (0,0)

Graphing Relations
To graph the relation in the previous example:
Write as ordered pairs (-3,3), (1,-2), (1,1), (4,4)
Plot the points

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Second level
● Third level
● Fourth level
● Fifth level
(-3,3) (4,4)
(1,1)
(1,-2)

Same with the points (-3,3), (1,1),
(3,1), (4,-2)

Second level
● Third level
● Fourth level
● Fifth level
(-3,3)
(4,-2)
(1,1)
(3,1)

Vertical Line Test
You can use the vertical line test to visually
determine if a relation is a function.
Slide any vertical line (pencil) across the graph to
see if any two points lie on the same vertical line.
If there are not two points on the same vertical line
then the relation is a function.
If there are two points on the same vertical line
then the relation is NOT a function

Second level
● Third level
● Fourth level
● Fifth level
(-3,3)
(4,4)
(1,1)
(1,-2)
Use the vertical line test to visually check if the
Function?
No, Two points are on
The same vertical line.

Second level
● Third level
● Fourth level
● Fifth level
(-3,3)
(4,-2)
(1,1)
(3,1)
Use the vertical line test to visually check if the
Function?
Yes, no two points are
on the same vertical lin

2. {(4, 1), (5, 2), (5, 3), (6, 6), (1, 9)}
Determine whether the relation is a
function.
f(x)
4 1
f(x)
5 2
f(x)
5 3
f(x)
6 6
f(x)
1 9
NO,
5 is paired with 2 numbers!

Is this relation a function?
{(1,3), (2,3), (3,3)}
1. Yes
2. No
Answer Now

Vertical Line Test (pencil test)
If any vertical line passes through more than
one point of the graph, then that relation is
not a function.
Are these functions?
FUNCTION! FUNCTION! NO!

Vertical Line Test
NO!
FUNCTION!
FUNCTION!
NO!

Is this a graph of a function?
1. Yes
2. No
Answer Now

Given f(x) = 3x - 2, find:
1) f(3)
2) f(-2)
3(3)-23 7
3(-2)-2-2 -8
= 7
= -8

Given h(z) = z2 - 4z + 9, find h(-3)
(-3)2-4(-3)+9-3 30
9 + 12 + 9
h(-3) = 30

Given g(x) = x2 – 2, find g(4)
Answer Now
1. 2
2. 6
3. 14
4. 18

To determine if a relation is a function

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