Unit 4
Functions & Relations
Write down at least ten relationships that exist
in the real world. Look for connections.
Ordered Pairs
• We can make ordered pairs out of the things
on the warm-up.
• (getting rid of headache, aspirin)
What do I need to write?
• Write the notes that are in all black, bold and
underlined.
In math, a relation is just a set of
ordered pairs.
• { (0,1) , (-3,4), (3,-2) }
• { (0, 1) , (5, 2), (-3, -1) }
• { (-1,7) , (1, 7), (3, 7), (2, 7) }
• A relation is just a set of ordered pairs. There
is absolutely nothing special at all about the
numbers that are in a relation. In other
words, any bunch of numbers is a relation so
long as these numbers come in pairs.
The domain and range of a relation
• The domain is the set of all the first numbers of the
ordered pairs .
• The range is the set of the second numbers in each
pair, or the y-values.
(4,5)
Domain
Input
Independent Variable
Range
Output
Dependent Variable
What’s the difference between a relation
and a function?
• In mathematics, what distinguishes a
function from a relation is that each x value
in a function has one and only ONE y-value.
• What does that mean?
Which relations below are functions?
• Relation #1 { (-1,2), (-4,51), (1,2), (8,-51) }
Relation #2 { (13,14), (13,5) , (16,7), (18,13) }
Relation #3 { (3,90), (4,54), (6,71), (8,90) }
• Relation #1 and Relation #3 are both
functions.
• Write an example of set of ordered pair that
is a function and a non-function.
Which relations below are functions?
• Relation #1 { (3,4), (4,5), (6,7), (8,9) }
Relation #2 { (3,4), (4,5), (6,7), (3,9) }
Relation #3 { (-3,4), (4,-5), (0,0), (8,9) }
Relation #4 { (8, 11), (34,5), (6,17), (8,19) }
• Relation #1 and Relation #3 are functions
because each x value, each element in the
domain, has one and only one y value, or one
and only number in the range.
For the following relation to be a function,
X can not be what values?
• { (8, 11), (34,5), (6,17), (X ,22) }
• X cannot be 8, 34, or 6.
Find the domain and range of the relation. Also,
determine whether the relation is a function.
Reading the ordered pairs
off of the graph, we get
{(2, 3), (2, 4), (3, 3), (4, 3)}
Also, determine whether the relation
is a function.
Reading the ordered pairs
off of the diagram we get
{(a, 1), (b, 2), (c, 1), (d, 2)}
•Look at the mapping diagram
•A function does NOT have an input that repeats.
•In other words, two arrows cannot come from one input.
•It is OK to have more than one arrow going to an OUTPUT.
•Draw a mapping diagram of a function and non-function.
We can also determine a function with
a T-chart.
y=x+7
•Plug in a number into x (domain,
independent variable, input) in
the equation.
•The answer that you get will be
the y (range, dependent variable,
output) of the equation.
•This is a function.
•Write a T-chart for a function
and non-function.
x
y
1
3
4
8
8
10
11
15
The Vertical Line Test
• The vertical Line test is a way to determine whether or not a
relation is a function. The vertical line test simply states that
if a vertical line intersects the relation's graph in more than
one place, then the relation is a NOT a function.
Are either of these functions?
Hint: Vertical Line Test
Draw a graph of a function and non-function.
Function Equations
• Function equation are almost always written
in a certain way. You will have y and x in
almost every equation, but they will be on
opposite sides of the equation.
y = equation with an x
y = 2x + 5
y = 5 horizontal (side to side) line across the y axis at 5
x = 2 vertical (up and down) line across the x axis at 2
http://score.kings.k12.ca.us/lessons/functions/machine.html
A vertical line (up and down) does NOT pass the
vertical line test. Ex. x = 15
Write an equation of a function and non-function.
Guess what a linear function is?
• Basically, a linear function is any straight line on a
graph except a vertical (up and down) line.
• Mathematically, a linear function is a relation that
has a pattern to it where the ordered pairs end up in
a straight line.
• The only straight line that is not a function is a
straight line.
Linear Function
Ticket out the door before you leave.
• Homework – Definition of a Function