Composition of Functions Video Lecture | Mathematics (Maths) Class 12 - JEE

hi everyone I am Nina from
mathematically inclined and today we are
discussing composition of functions the
entire class 12 syllabus has been
uploaded under separate playlists the
links for the same are in the
description box below please note all
these videos are crafted not only to
suit us abs he needs but also to help
you for the competitive exams so let's
get started in order to understand the
composition of functions I have taken
three sets in front of me name them as a
B and C now if we take a function f
which is moving from A to B and then
beginning with the stage set B I take
another function G from B to C
particular developments here let's say X
after applying the function f it becomes
f of X now beginning with f of X the
physics of the elements to my new
function G and gives us the final
product as G of FX in short the new
function which moves from A to C is
defined by G of F X so to define the
function we say F is a function from A
to B and G is the function from B to C
then G of F is defined from A to C as G
of FX means G applied to FX for all X
belonging to a you know it is like
considering a series of machines in a
factory but so once the raw material egg
is put into the first machine it takes
it to the next level and creates FX now
taking this as the input for the next
machine under the mapping G it creates
the final
which is G of FX it's more like a chain
reaction there are two very important
points to be remembered so if F is a
function from A to B and G is a function
from B to C range of F which is B should
be the starting point of the domain or
the next constant G so G of F 2 exists
likewise for F of G 2 exists the range
of G should be a subset of domain of F
that it brings us to the conclusion the
ranger will F of G & G of F they are not
the same
please remember for G of F F is a free
youth first followed by three and four F
of G G is operated first followed by F
so looking at the very first question
your function f is defined from reel to
reels and FX equal to X square either so
the real function defined this way we
have to find f of g and g of f beginning
with F of G let me take it it is the
same as being apply F to your G X in
place of G H we have 2x plus 1 now if f
of X is X square then F of 2x plus 1
becomes 2x plus 1 whole square similarly
talking of zero of FX we would be
applying G to F X so this gives us G of
X square if G of X is 2x plus 1 then G
of x squared or domain typically becomes
2x squared plus 1 and as you can clearly
see your f of G & G of F they give us
different values
second question is a very interesting
one it's well test you for a lot of
things so f and
have been defined in terms of ordered
pairs we need to write therefore GM G of
F but before we do that we need to at
least have these SNC's sorted so from
your previous capsules you know that the
domain of the function is the first
coordinates so domain of F over here is
or the first coordinate that is one
three and four and range of s by the
second coordinates which are two five
and one similarly domain of G would be
two five and one and range of G would be
three one these are the domain and range
now to rewrite this in the function form
this means F of 1 is 2 f of 3 is 5 and F
of 4 is 1 similarly G of 2 is 3 G of 5
is 1 and G of added 3 in order to begin
with F of G we start with whatever
elements are there in domain of G so we
need to find f of G of 2 F of G of 5 and
F of G of 1 which means F applied to G
of 2 G of 2 from here is 3 so it is the
same as saying F applied to 3 now F
applied to three is not the answer of 5
so f of G of 2 becomes 5 similarly
starting with F of G of 5 we say this is
f applied to G of 5 now G of 5 is just 1
and F of 1 further gives us the answer
to
continuing with the third part F of G of
1 which gives us G of 1 is 3 so f of 3
further gives us the answer as 5
so this means my F of G finally gives
must be dancers will do I have five with
five I get two and with one we get five
again this is the awesome lightweight
beginning with now G of F we take these
elements from the domain that means we
need to find what is G of F of one what
is G of F of 3 and G of F of four I
would request you all to pause the video
for a moment we'll try these on your own
if you have tried on your own please see
this is once again G of F of 1 f of 1 is
2 so G of 2 which finally is the 3
similarly the other ones and thus we say
G of F gives us the ordered pairs which
are 1 3 3 1 4 3 so these are the answers
now for question 3 you are given only
one function are the wire which is
defined as a explain minus 3 X plus 2
and you have to find the composition of
function with itself so that means for f
of F X we need to find f applied to FX
again which means F applied to X square
minus 3x plus 2 so so just to understand
let's say I call this as capital X so f
of capital X would give us capital X
square minus 3x plus 2 and now putting
back the value for this capital X you
get x squared minus 3x plus 2 whole
square minus twice of H square minus 3x
plus 2 plus 2 on simplifying all this
finally our F of FX becomes this so this
brings us to the end of composition
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more videos until then bye bye