L6 : Graphs of functions - Relations and Function, Mathematics, Class 11 Video Lecture

hello friends this video on relation and
functions part six is brought you by
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before watching this video you should
have watched relation functions part 1 -
part 5 let us draw some graph now graph
identity function
it says FX is written X domain and range
both are real numbers and the graph is
us let's draw this so when you say if it
is going to X you are saying y is equal
to X now let me draw the value here for
X is equal to 1 Y is going to 1 for X
square root 2 y is going to or x equal
to 3 y is 3 4 I split to minus 1 y is
minus 1 for ice will be 0 Y is let us
plot this values in the number line X is
equal to 1 Y is 1 X is equal to 2 y is 2
X equal to 3 y is 3 x squared - 4 y is 4
X equal to minus 1 Y is minus 1 x
squared 2 minus 2 y is - do you see this
is 1 comma 1 this is 2 comma 2 because X
is 2 here Y is 2 air this is again 3
comma 3 and this is 4 comma 4 you draw
this graph like this you get a graph
like this straight line not even draw
straight line but it is straight line
and this is called identity function so
whatever is the value of x the same is
the value of y so it's identity function
in the graph is a straight line that
domain and range both are are so the x
axis is the domain and the y axis is the
range so why is range and X is domain
see X can take any real number and y can
take any real number now let us take
constant function in constant function
we say FX is going to K here domain is
our and range is only K and
parallel line parallel to x axis let's
take an example and we say FX is equal
to four so you say why and usually X you
give any value of x you say X equal to 1
the Y is 4 you give x squared - 2 y is 4
you give X go to 0 Y is 4
you gave 3 y is 4 you gave minus 2 y is
so for any value of x Y is always 4 I'll
plot this where you give X 0 y is 4 when
you give 2 y is 4 when I give for wise
more when you give minus 2 y is 4 minus
4 y is 4 for any number y is 4 so this
is a straight line parallel to X so if
you see here the domain that is x axis
this are any real number can take this
value but the range you see the range is
4 because 4 is the only order so in this
case domain is our and ranges so and
this line parallel to x axis let us take
an example of polynomial function
polynomial function is a form a not X a
1x a not a 1 X into X square plus plus
plus a 1 X square X to the power n where
N is a non-negative integer and a 1 a 2
is rear of real numbers here domain is
our we can give any can give any real
number but the range depends on the
function of X and the graph is also it
depends on the function of X for example
let's take function FX is equal to take
function FX will go to this this is -
let me take this as one thing this is
one another as 0 should be 2 plus X plus
X square this is the function now let me
create Y in X now I give Y as X as 0
you give 0 Y is going to 2 plus 0 plus 0
you give X is equal to 1 what you get 2
plus 1 plus 1 that is 4
you have X equal to minus one what you
get is minus 1 plus 1 0 plus 2 - you
gave X is equal to 2 what you get is 2
plus 2 plus 2 square for that has 1 2 4
4 plus 2 6 plus 2 8 this is 8 you give
minus 1 you get minus 2 plus 2 0 plus 2
minus 2 square over 2
similarly you get 3 you get 3 square 9
plus 3 12 plus 2 14 and you give minus 3
you get minus 3 plus 2 minus 1 minus 1
plus 3 square 9 is going to a still get
something like this now let's plot this
points in the graph so when x is 0 Y is
2 when X is 1 Y is 4 when x is -1 y is 2
when X is 2 y is 8 when X is 3 y is 14
somewhere here when X is minus 3 y is 8
so matter and X is minus 2 y is 4 so you
plot this graph graph will come
something like this somewhere it will
touch 0 and we have this form correct
this is the graphical representation of
this watching so here you see the graphs
which you get it all depends on the
function we are taking so we say the
graph step into the function and the
range also depends on the function now
let's take rational function rational
function in the form of FX by GX where
GX is not equal to 0 obviously if you
make the X is equal to 0 it will become
infinity sample is FX is equal to 1 by X
and here also try
change domain and range would depend on
X in this case both domain and range
matrix because if I say 1 by X then X
can't be 0 there is a shoo-in domain
domain does not depend on X and if I say
1 by X minus 2 here also X can't be do
so the domain also depends on the
function range also depends on the
function let's draw this rough expert 2
1 by X so we'll say Y is equal to 1 by X
is X and we get Y so you say X go to 0 Y
becomes infinite 1 by 0 you give X go to
1 Y become 1 X go to minus 1 Y become
minus 1 takes go to Y become 1 by 2 x
squared - -2 y am combined as 1 by 2 X
go to 3 positive 3 y is equal to minus
plus 1 by 3 x squared 2 minus 3 y we go
minus 1 by its plot this X going to 0
its infinite we can brought us now X go
to 1 is 1 1 is 1 x / 2 minus 1 Y is
equal to minus 1 this is the point X is
equal to Y is going to 1 by 2 is going
to do Y is equal to less only less than
this correct and X square to minus 1 by
2 y is equal to minus 1 by 2 this
further less is here 4 3 to come
somewhere here from here X go to 0 Y is
equal to infinite is here in fact is
this positive possible infrared so we
will draw graph from here from infinite
it comes and it goes if you give - if
you give infinity it goes to 0 similarly
in this case when you give - if you give
0 y is going to again infinite correct
so Y is going to in pan aye - infrared
sorry not working correctly and here it
is bending like this because if you give
minus infinity X will devise ability why
become almost zero so it will - - or
Mizoram but if you give X is equal to
almost zero it becomes infinity this is
the graph of 1 by X again the graph
depends on function to function here in
this case domain range
everything depends on function now we'll
see models function modulus functions FX
go to FX module X if you give - 1 as
input you will get 1 if you have - 2 you
will get D so whatever value you provide
you get the whole step al in here domain
is are you can give any real number but
since it always give positive number the
ranges hardness will draw this will take
X and will take Y then we give one has
input the output is 1 when you give
minus 1 or plus 1 when I give two as
input the output is 2 and I give - - or
do this plus 2 any negative number you
give you get the pasta coffee let us
draw this IV gives 0 you get 0 output
you give plus 4 you get 4 1 let us draw
this 0 is 0 you give one you get one
output eta 2 we get 2 or 2 to give 3 you
get 3 outputting for you get for all but
they give minus 1 you get 1 out you give
- to get to as output
you give minus 3 you get 3 as output you
get 4 minus 4 get further out so the
graph is something like this this and
this so if you observe the output is
always positive so I say the range is
always positive or positive but that
domain the x-axis can be anything
positive or negative so it says domain
is any real number but the output that
is range is always positive so we can
see all this thing from the graphs also
that we have seen the made this graph
and from the graph we see X can take any
values from minus infinity plus infinity
so we will say the winner is why I can
take only positive value students ranges
are positive now in take graph of Signum
function so what is Signum function
Signum function is X is equal to 1 X is
greater than 0 X is equal to 0 if it is
exactly 0
and X equal to minus 1 if it is less
than 0 now let's let's create Y and X Y
is equal to FX here so you zero our
output again if you give one without put
ticket one you have to as output you get
2 so you get 1
if 3 is output you get 1 similarly if
you get minus 1 output you get minus 1
minus 2 again you are put is minus 1
give minus 3 as the input you get minus
1 as output because it says for any
number less than 0 it is minus 1 to draw
this for 0 it is 0 that is confirmed in
part is form for any number greater than
0 let's suppose 0.001 it is here it is
here 1 4 2 it is 1 before it is 1 / 3 it
is 1 all this points are there similarly
4 minus minus 1 it is 1 sorry 1 this is
cancelled minus 2 is 1 for everything it
is 1 but you see or 0.00 needs one if
you say is 0.0001 this is 1 but for 0 it
is not 1 so this is example of open
interval which we have studied and we
know our memory trip o-n-e that is open
know in points there is an end point is
not included so it is open interval and
open interval is notified a light box so
will create a light box here and here
also will create a light box and will
extend this line this is the graph of
signal function repeat Signum function
or any number greater than 0 it is 1 any
number less than 0 is minus 1 and for 0
it is 0 since for its point 0 0 0 1 also
it is 1 so it is example of open
interval and in open interval use white
box white circle so we have used the
white circle non a dark circle and we
have plotted the points this is an
example of Signum function graph here
domain is our and range is minus 1 0 & 1
you see
you can give any value as the the way
but the range is either 0 equals 0 or
minus 1 or 1 so from graph we can say
what is the domain and range thank you
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