Relations & Functions Video Lecture | Additional Study Material for Class 12

in this video we're going to take a look
at relations and functions relations and
functions remember functions are things
where we have no repeating x-values are
there's no repeating values in the
domain we can test that by looking at
the ordered pairs or if we have a graph
we can use the vertical line test
we'll look at the vertical line test a
little bit later but let's start by
looking at some sets of ordered pairs
and determine whether or not they're
functions all right this first one if we
look we have 2 5 0 and for all those X
values are different so this would be a
function we do have repeat our repeating
y-value which is the 3 but that's okay
so in this case yes this is a function
the domain of this situation we just
pick out the x value so let's just go
ahead and list those out and usually
we'll put them in order so we start with
0 now we have 2 we have 4 & 5 so the
domain in that function is 0 2 4 & 5 now
the range is the set of Y values so the
Y's we have 3 4 3 & 1
I'm going to put them in order and if
they show up more than once
I only have to write them once so we
don't have to write the 3 twice even
though it is in our ordered pairs twice
okay finally to make a mapping diagram
what we do is we put each the domain and
the range into sort of a little bubble
here so our domain is 0 2 4 & 5 our
range is 1 3 & 4 and then we're just
going to draw arrows to connect the
ordered pairs so our first one
is two to three so two to three
our next one is five to four just drawn
arrows again zero to three so here we've
got zero we're going to three again and
then we've got four to one so that would
be our mapping diagram for that relation
let's take a look at another one right
here is this a function well if we're
given ordered pairs remember we look at
those x-coordinates so we've got a 3 a 2
a 0 and a 2 oh we've got to repeat when
we repeat in our X's it is not a
function this is not a function we can
however still pick out the domain and
range the domain is the set of X values
again so we have 0 2 & 3 and our range
is the set of the Y values we have
negative 1 negative 2 2 & 1 I'm just
going to put those in order negative 2
negative 1 1 & 2 okay and if we want to
make a mapping diagram there just make
our little bubbles put those numbers
inside our domain is 0 2 & 3 our range
is negative 2 negative 1 1 & 2 and then
I'm just going to switch colors here so
that we can maybe see this a little
better and if we then just connect our
what our ordered pairs are so 3 is being
paired with negative 1 like so 2 is
going with negative 2 so we're up there
0 is going with 2 and finally 2 is going
with 1 okay so there's our mapping
diagram for that relation now the
vertical line test switch
mentioned earlier we can use that also
to determine whether a relation is a
function so here's a couple examples of
graphs and if we look at them and use
the vertical line test the vertical line
test says that if a vertical line
passing over the graphed touches the
graph no more than once then that is a
function so in this case as I take my
vertical line over across my graph well
it's just touching once touching once
touching once touching once and no more
so this first one would be yes that is a
function this second one as I'm going
along over here I'm okay because it's
not touching at all it can still be a
function if my vertical line is not
touching it I get over here to the
y-axis I'm touching at one point that's
okay here I'm touching at one point one
point oh three points oh oh that's
trouble so right here is the problem
we're touching at three points this
graph fails the vertical line test
because it's touching more than once
so this one is not a function so again
this first one is a function because it
passes the vertical line test that
touches at most once and the second one
is not a function because it fails the
vertical line test right here let's look
at two more quickly again our vertical
line test here's my vertical line coming
along coming along I'm okay I'm okay I'm
okay I'm okay sometimes people might
have a little bit of a question here but
notice how it's just one point at three
zero that is touching so we're okay
we're okay this first one we indeed pass
the vertical line test how about this
one right here we're okay okay okay okay
we got trouble right here we're hitting
three times right there so this fails
the vertical line test in this whole
area because we're touching the graph
again is touching our vertical line more
than once right over here it's okay we
just be touching once here but this area
right here is trouble so our vertical
line test is failed on this second one
so again this first one yes it is indeed
a function passes the vertical line test
this second one no it is not because it
fails the vertical line test hope this
was helpful we can use both the vertical
line test and by looking at the X values
if we're given ordered pairs or maybe a
table to determine if a relation is a
function