in this video we're going to focus on
relations and functions
so what is a relation
a relation is a set of pairs of input
and output values
here we have three ordered pairs
in the first relation on the left
the x value
is the input value
the y value is the output value
the x values is associated with the
domain of the relation the y values is
associated with
the range of the relation
so now let's focus on part a list the
domain and range of each relation
so let's start with the domain
so what we're going to do is we're going
to make a list of all of the x values
and i'm going to write it in ascendant
order
so first we have negative three
and then zero
and two
now let's write the range
of that relation
so we're going to focus on
the y values and it's already listed in
ascendant order
so 1 4
and 5.
now let's do the same thing for the
other relation
so let's write out the domain
so the lowest
x value is negative two
next is one and then three
now let's write out the range
of that relation
the lowest y value
is negative two
and then it's
three four and seven
so that's how you can write out the
domain and range of each relation
now how can we determine if the relation
is a function
in order for the relation to be a
function
every input value
must have
only one output value
if an input value corresponds to two or
more output values
that relation is not a function
now
let's focus on the first relation
so we have the ordered pair 2 1 the
input value is 2 the output value is 1.
and then negative 3 4. so negative 3
corresponds to 4
and then 0 corresponds to 5.
so for the first relation we can see
that
for every input value there's only one
output value
now let's focus on the second relation
so we have the ordered pair one comma
three
next is negative two four
and then it's three negative two
and then finally negative two seven
so for the second relation notice that
negative two corresponds
to two different output values
now that's a problem
if you put in an input value of negative
two should the output be four or seven
so whenever you have that situation you
know that relation
is not
a function
the first one is a function
every input value corresponds to an
output value just one output value
so a quick way to
look at a relation to see if it's
if it's not going to be a function
look for repeating x values
if you see
two x values that are the same
but correspond to two different y values
then you know the relation is not a
function
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and let's get back to the video
now let's move on to the next example
draw a mapping diagram of each relation
shown below
so let's start with the relation on the
left
we're going to map out the domain and
arrange
so for the domain we have the values
negative 2
1 and 3.
for the range we have the y values
negative 6
0
and 4.
now negative two
corresponds to zero
one corresponds to four
three
corresponds to negative six
so for every input value on the domain
side there's one corresponding
output value on a range side
so this
is a relation
i mean
this relation is a function so the
answer is yes for
the first relation
now let's move on to the second
relation
so let's create a mapping diagram as
well
so let's start with the domain
the lowest x value is negative two
next we have
zero
and then the last one is three
now looking at the y values
the lowest one is negative one
and then it's going to be one
two and five
so negative two corresponds
to positive one
zero corresponds to five
three corresponds to two
and zero
corresponds to negative one
so just by seeing
the repeat x values that we see here we
could tell that
this is not going to be
a function
the two x values have two different y
values
you can see zero points to negative one
and five
so the second relation
is not
a function
now for this one what we're going to do
is we're going to draw a function table
of the relation
and then we're going to determine if the
relation is a function
so in this table we're going to list
the input values
next to the output values
the input values represent the x values
the output values represent the y values
so the input values it corresponds to
the domain and the output values
corresponds to the range
so the lowest input value that we have
is negative three
next
is one
and then we have another one
and then after that is
it's three and five
now for this function table
i'm going to write the input value twice
because that's what we have here
when writing out
the domain and the range
for repeat values we would write repeat
values once
now negative three corresponds to two
for the table
these numbers need to match so i'm not
going to list the output values in
ascendant order
now for this one
we could use either one so i'm going to
use 1 2 for the next one and then 1 4.
now when x is 3 y is 7
and when x is 5 y is negative 4.
so that's the function table
and because we have
two identical x values that correspond
to two different y values
we know that this relation
is not
a function
so that's it for this problem
when you have a graph
the best way to determine if the graph
represents a function is to use the
vertical line test and that's what we're
going to do in this problem
so any way you draw a vertical line
for the first graph notice that
the line only touches the graph at one
point
therefore
this is the answer is yes it represents
a function
for the next one on the right if we draw
a vertical line
notice at
this point or place a line at that
location
we have
three points of intersection between a
graph and a vertical line
if we can get two or more points on a
vertical line then a relation is not a
function so we're going to say no
now if we put the vertical line here
notice that we have five points on that
line
so this relation is not a function
for the next relation
it doesn't matter where we put the graph
we will only get
i mean doesn't matter where we put the
vertical line we're only going to get
one point
if we put it here
it's only going to touch the line once
so we can't draw a vertical line where
it touches two points therefore
this relation represents a function
for the circle
if we put the line here
we can get two points of intersection
so we're going to say yes i mean no not
yes
this is no
the circle does not represent a function
it does not pass the vertical line test
it touches the line
at two points in order for it to pass
the vertical line test the graph must
touch the line only at one point
as we
saw in these two cases
so that's how you can use the vertical
line test to determine
if a relation represented by a graph
is a function