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Algebra of functions and examples - Relations & Functions Video Lecture - Class 11
FAQs on Algebra of functions and examples - Relations & Functions Video Lecture - Class 11
| 1. What is the algebra of functions? |  |
Ans. The algebra of functions refers to the set of rules and operations that can be performed on functions. It involves combining, manipulating, and simplifying functions using operations such as addition, subtraction, multiplication, and division.
| 2. How do you add or subtract functions? | |
Ans. To add or subtract functions, you simply add or subtract the corresponding terms of the functions. For example, if you have two functions f(x) and g(x), their sum (f+g)(x) is obtained by adding the values of f(x) and g(x) at each point x. Similarly, the difference (f-g)(x) is obtained by subtracting the values of g(x) from f(x) at each point x.
| 3. Can you provide an example of multiplying functions? | |
Ans. Yes, let's consider two functions f(x) = 2x and g(x) = x^2. To multiply these functions, you multiply the corresponding terms. So, (f * g)(x) = (2x)(x^2) = 2x^3. This means that the product of f(x) and g(x) is a new function that is equal to 2x^3.
| 4. How do you divide functions? | |
Ans. Division of functions is done by dividing the corresponding terms of the functions. For example, if you have two functions f(x) and g(x), their quotient (f/g)(x) is obtained by dividing the values of f(x) by g(x) at each point x. It is important to note that division by zero is undefined, so you need to ensure that g(x) is not equal to zero.
| 5. Can you simplify functions using algebraic operations? | |
Ans. Yes, algebraic operations can be used to simplify functions. For example, you can simplify expressions involving functions by combining like terms, factoring, or applying rules of exponents. These simplifications can help in analyzing and understanding the behavior of functions.
Text Transcript from Video
hello friends this video on relation and
function part 7 is brought to you by
exam fear calm no more fear from exam
before watching this video you should
have watched relation and functions part
1 to part 6 the graph of greatest
integer function so what is greatest
integer function greatest integer
function it takes the greatest value it
is denoted by this box in X and in this
case domain is on and real range is only
the whole number because if you give one
point to it will make you it will make
it two if you give one point seven do
also make it do so range is only the
whole number so the definition is if the
number is between minus one and zero it
becomes minus 1 the number is between 0
& 1 it becomes zero if the numbers
between 1 & 2 because 1 sorry in this
one point will become 1 and 1 point 7
become 1 so let us take the same y and
this is X till say X is equal to X cut
to width say 1 output is 1 we will say 1
point to output is 1 correct because
this is the bracket machine line they
say 5 point 2 dot will be fine they say
4.20 momentary it will be 4 at 6 in the
interval also will say minus 1 point 5
so minus 1 point 5 is not in this range
so to become minus 2 you will say minus
0.5 it will become minus 1 now let's
plot this point well take so X is equal
to 0 Y is equal to 0 so 0 0 is 1 point
here then you say X is equal to 1 Y is
equal to 1 when you say X equal to 1
point 2 y is equal to 1 X equal to 5
point 2 y is equal to
5x is going to do 4.3 y is equal to 4x X
go to minus 2 minus 1.5 this range Y is
equal to minus 2 X is equal to in this
range 0 & 1 it is minus 1 so if you see
this function for any value from 0 to 1
the answer is 0 similarly for 1 to 2 is
2 2 to 3 3 to 4 3 4 to 5 for 5 to 6 5 C
to 7 6 similarly 0 to minus 1 minus 1
minus 1 2 minus 2 minus 2 minus 2 2
minus 3 minus 3 so this is how the
Signum function looks like also going to
be noted that if it is 1 it is not 2 is
1 it is 1 you know that if it is 1 it is
not equal to 0 it is 1 so here we'll put
a open interval why because for 1 this
is 1 not 0 here also will put open
interval similarly we will have to put
open interval everywhere on the tip
because at that point of time it takes
the higher value for example in this
case for 6 it does not fight for what
for you say 0.5 point 9 9 9 is 5 but the
moment is 6 then this value will be not
counted so in that case this value will
be taken so on the tip we have to create
this open intervals this is mandatory if
you don't do this the graph is wrong
without open interval this graph is
wrong please make a note of this open
interval is very important there because
example when you say 4 I mean you say 4
point 9 9 9 9 this will give you 4 but
the moment you say 5 this will you fine
so here in this case 5
say four point nine nine nine it is this
M is a five it is five so 0.9 and n is
given by open interval of this now let
us discuss algebra of functions so we
can do we can add two real functions we
can subtract two real functions you can
multiply a function by a scalar you can
multiply two real functions we can
divide two real functions where we have
to make sure that GX the is not zero
because if it 0 FX by ZX it becomes
impolite let's take a tissue
so when you say if X 1 to F FX plus GX
let's say FX is equal to 2x plus 3 and
GX is equal to X square plus 3 X so when
you say FX plus GX you add these two
functions this becomes 2x plus 3 plus X
square plus 3x it becomes x squared plus
5x plus 3 so this is one example of
addition of two real functions let us
take subtraction of the real function FX
minus GX similarly we can have FX is
equal to X square plus 2x plus 1 and GX
is going to do X plus 8 so when you say
FX minus GX you actually subtract these
two so what you get is X square plus 2x
minus X plus 1 minus 8 this is nothing
but x squared minus X minus self this is
FX minus 2 G X similarly when you say
that you multiply a scalar with a
function so for example you have FX is
equal to X square plus 2x plus 3 and you
have a is equal to 5 then you say a into
FX or 5
into FX again 2x squared plus 2x plus 3
this is equal to 5x squared plus 10x
plus 50 when we say that we multiply two
function then we actually multiply FX
equal to let's say X plus 1 and FY is
going to 2x plus 3 then we say like this
GX then you say FX into GX you mean to
say X plus 1 into 2 X plus 3 and that is
equal to 2x squared plus 3x plus 2x plus
3 and this is equal to 2x squared plus
5x plus 3 this is the answer so you can
multiply two functions similarly you can
divide two functionals for example where
FX is equal to X cubed plus 2x squared
plus 3x and you have GX is equal to X
you say FX by GX this is equal to X
cubed plus 2x squared plus 3x divided by
X and this is equal to X square plus 2x
plus thus we have seen that we can add
two function and subtract the functions
we can divide two functions we can
multiply two functions or we can
multiply a scalar to a function now we
have a set given in form of X comma Y we
have to find linear equation linear
equation is always in the form Y is
going to MX plus C correct so now we
have been given the X is equal to 1 Y is
equal to 1 we get x squared - 2 y is
equal to 3
if you give X equal to zero Y is equal
to minus one like that so let's first
put X go to 0 so in this case this will
be 0 we will say minus 1 is equal to M
into X 0 is equal to 0 plus C so C is
equal to minus 1 that is clear now so we
have equation or form y is equal to MX
minus 1 now let me put X getting 1 Y it
is not put as one so say 1 is equal to M
minus 1 so M is equal to 2 that is also
clear so our equation is actually in the
form of Y is equal to 2x minus 1 so now
we have to find the function so the
function is nothing but FX equal to 2x
minus 1 this is the what we have done we
have taken the general equation FX is
equal to MX plus C or Y is build MX plus
C because that is a linear equation they
are talking about so now and they have
given different values so we have put
the values and found the value of M n X
m and C M comes out to be 2 and C comes
out minus 1 so we are replace the value
of M and C in this equation so we got Y
is going to 2x minus 1 if x squared X
minus 1 is the answer here we have a
question where we want to find that
domain of function so we have this is of
the form FX by GX now in this case G X
should not be equal to 0 because if it
is equal to 0 this will become infinite
that means X square minus 5x plus 4 the
one in the denominator should not be
equal to 7 so should not be equal to Z
so we say X square minus 4x minus X plus
4 Shimon because 0
or X minus 4 into X minus 1 should not
be 0 or X should not be equal to 4 comma
1 so when we say when we are talking
about domain domain is any our real
number minus 4 comma 1 Center and this
is the answer you got what we had done
so we have seen that this was of the
form FX by GX and we know that this
function to exist GH should not be xx 0
and that's what we have done we have
don't let X square minus 5x plus 4 is
greater would not be equal to 0 we have
solved this and we form that X should
not be equal to 4 or 1 so then the
weight of this function is any real
number
these are any real number - for gamma 1
let's say one way example where FX is
going to this n GX will do this we have
to see with this function or not so for
0 to 3 they say will write the value 0 1
2 3 4 5 6 7 8 9 4 0 2 3 it is x squared
it is 0 Square 1 square 2 - square 3
Square on 3 to 10 it is 3 X 3 into 3 is
9 9 3 to 4 12 315 16 18 21 83 24 93 d7
and 3 in 10 13 so this is the relation
this is X and this is f F X so we see
that for every input there is only one
output country we have only 9 before we
have 12 v EF happening so it is a
function now let's see the scenario 2
again let me write 0 1 2 3 4 5 6 7 8 9
10 for 0 to 2 it is X square 0 square 1
square 2 squared 3 squared so are
equally 0 to 2 square now from 2 to 10
it is 3 X so R 2 also it is 2 into 3 6
which is trained to 2
which is 3 into 3 9 4 into 3 12 15 5
into 3
63 1873 2183 2419 to 327 and three into
ten thirty so we see for a given input -
they are 2 or 4 n shapes so this is not
a function but this is a function so
what we have done we found all the
values of x and y and we found that in
first case for every input there's only
one output but for second case GX we
found it forever - they are both so GX
is a function
thank you visit exam viacom to watch
free education videos try free online
tests in the best quality study material
study from the best tutor and mentors
function part 7 is brought to you by
exam fear calm no more fear from exam
before watching this video you should
have watched relation and functions part
1 to part 6 the graph of greatest
integer function so what is greatest
integer function greatest integer
function it takes the greatest value it
is denoted by this box in X and in this
case domain is on and real range is only
the whole number because if you give one
point to it will make you it will make
it two if you give one point seven do
also make it do so range is only the
whole number so the definition is if the
number is between minus one and zero it
becomes minus 1 the number is between 0
& 1 it becomes zero if the numbers
between 1 & 2 because 1 sorry in this
one point will become 1 and 1 point 7
become 1 so let us take the same y and
this is X till say X is equal to X cut
to width say 1 output is 1 we will say 1
point to output is 1 correct because
this is the bracket machine line they
say 5 point 2 dot will be fine they say
4.20 momentary it will be 4 at 6 in the
interval also will say minus 1 point 5
so minus 1 point 5 is not in this range
so to become minus 2 you will say minus
0.5 it will become minus 1 now let's
plot this point well take so X is equal
to 0 Y is equal to 0 so 0 0 is 1 point
here then you say X is equal to 1 Y is
equal to 1 when you say X equal to 1
point 2 y is equal to 1 X equal to 5
point 2 y is equal to
5x is going to do 4.3 y is equal to 4x X
go to minus 2 minus 1.5 this range Y is
equal to minus 2 X is equal to in this
range 0 & 1 it is minus 1 so if you see
this function for any value from 0 to 1
the answer is 0 similarly for 1 to 2 is
2 2 to 3 3 to 4 3 4 to 5 for 5 to 6 5 C
to 7 6 similarly 0 to minus 1 minus 1
minus 1 2 minus 2 minus 2 minus 2 2
minus 3 minus 3 so this is how the
Signum function looks like also going to
be noted that if it is 1 it is not 2 is
1 it is 1 you know that if it is 1 it is
not equal to 0 it is 1 so here we'll put
a open interval why because for 1 this
is 1 not 0 here also will put open
interval similarly we will have to put
open interval everywhere on the tip
because at that point of time it takes
the higher value for example in this
case for 6 it does not fight for what
for you say 0.5 point 9 9 9 is 5 but the
moment is 6 then this value will be not
counted so in that case this value will
be taken so on the tip we have to create
this open intervals this is mandatory if
you don't do this the graph is wrong
without open interval this graph is
wrong please make a note of this open
interval is very important there because
example when you say 4 I mean you say 4
point 9 9 9 9 this will give you 4 but
the moment you say 5 this will you fine
so here in this case 5
say four point nine nine nine it is this
M is a five it is five so 0.9 and n is
given by open interval of this now let
us discuss algebra of functions so we
can do we can add two real functions we
can subtract two real functions you can
multiply a function by a scalar you can
multiply two real functions we can
divide two real functions where we have
to make sure that GX the is not zero
because if it 0 FX by ZX it becomes
impolite let's take a tissue
so when you say if X 1 to F FX plus GX
let's say FX is equal to 2x plus 3 and
GX is equal to X square plus 3 X so when
you say FX plus GX you add these two
functions this becomes 2x plus 3 plus X
square plus 3x it becomes x squared plus
5x plus 3 so this is one example of
addition of two real functions let us
take subtraction of the real function FX
minus GX similarly we can have FX is
equal to X square plus 2x plus 1 and GX
is going to do X plus 8 so when you say
FX minus GX you actually subtract these
two so what you get is X square plus 2x
minus X plus 1 minus 8 this is nothing
but x squared minus X minus self this is
FX minus 2 G X similarly when you say
that you multiply a scalar with a
function so for example you have FX is
equal to X square plus 2x plus 3 and you
have a is equal to 5 then you say a into
FX or 5
into FX again 2x squared plus 2x plus 3
this is equal to 5x squared plus 10x
plus 50 when we say that we multiply two
function then we actually multiply FX
equal to let's say X plus 1 and FY is
going to 2x plus 3 then we say like this
GX then you say FX into GX you mean to
say X plus 1 into 2 X plus 3 and that is
equal to 2x squared plus 3x plus 2x plus
3 and this is equal to 2x squared plus
5x plus 3 this is the answer so you can
multiply two functions similarly you can
divide two functionals for example where
FX is equal to X cubed plus 2x squared
plus 3x and you have GX is equal to X
you say FX by GX this is equal to X
cubed plus 2x squared plus 3x divided by
X and this is equal to X square plus 2x
plus thus we have seen that we can add
two function and subtract the functions
we can divide two functions we can
multiply two functions or we can
multiply a scalar to a function now we
have a set given in form of X comma Y we
have to find linear equation linear
equation is always in the form Y is
going to MX plus C correct so now we
have been given the X is equal to 1 Y is
equal to 1 we get x squared - 2 y is
equal to 3
if you give X equal to zero Y is equal
to minus one like that so let's first
put X go to 0 so in this case this will
be 0 we will say minus 1 is equal to M
into X 0 is equal to 0 plus C so C is
equal to minus 1 that is clear now so we
have equation or form y is equal to MX
minus 1 now let me put X getting 1 Y it
is not put as one so say 1 is equal to M
minus 1 so M is equal to 2 that is also
clear so our equation is actually in the
form of Y is equal to 2x minus 1 so now
we have to find the function so the
function is nothing but FX equal to 2x
minus 1 this is the what we have done we
have taken the general equation FX is
equal to MX plus C or Y is build MX plus
C because that is a linear equation they
are talking about so now and they have
given different values so we have put
the values and found the value of M n X
m and C M comes out to be 2 and C comes
out minus 1 so we are replace the value
of M and C in this equation so we got Y
is going to 2x minus 1 if x squared X
minus 1 is the answer here we have a
question where we want to find that
domain of function so we have this is of
the form FX by GX now in this case G X
should not be equal to 0 because if it
is equal to 0 this will become infinite
that means X square minus 5x plus 4 the
one in the denominator should not be
equal to 7 so should not be equal to Z
so we say X square minus 4x minus X plus
4 Shimon because 0
or X minus 4 into X minus 1 should not
be 0 or X should not be equal to 4 comma
1 so when we say when we are talking
about domain domain is any our real
number minus 4 comma 1 Center and this
is the answer you got what we had done
so we have seen that this was of the
form FX by GX and we know that this
function to exist GH should not be xx 0
and that's what we have done we have
don't let X square minus 5x plus 4 is
greater would not be equal to 0 we have
solved this and we form that X should
not be equal to 4 or 1 so then the
weight of this function is any real
number
these are any real number - for gamma 1
let's say one way example where FX is
going to this n GX will do this we have
to see with this function or not so for
0 to 3 they say will write the value 0 1
2 3 4 5 6 7 8 9 4 0 2 3 it is x squared
it is 0 Square 1 square 2 - square 3
Square on 3 to 10 it is 3 X 3 into 3 is
9 9 3 to 4 12 315 16 18 21 83 24 93 d7
and 3 in 10 13 so this is the relation
this is X and this is f F X so we see
that for every input there is only one
output country we have only 9 before we
have 12 v EF happening so it is a
function now let's see the scenario 2
again let me write 0 1 2 3 4 5 6 7 8 9
10 for 0 to 2 it is X square 0 square 1
square 2 squared 3 squared so are
equally 0 to 2 square now from 2 to 10
it is 3 X so R 2 also it is 2 into 3 6
which is trained to 2
which is 3 into 3 9 4 into 3 12 15 5
into 3
63 1873 2183 2419 to 327 and three into
ten thirty so we see for a given input -
they are 2 or 4 n shapes so this is not
a function but this is a function so
what we have done we found all the
values of x and y and we found that in
first case for every input there's only
one output but for second case GX we
found it forever - they are both so GX
is a function
thank you visit exam viacom to watch
free education videos try free online
tests in the best quality study material
study from the best tutor and mentors
About this Video
Oct 01, 2025
Last updated
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