Class 12 Exam > Class 12 Videos > Example Invertible functions-3 (Part - 34) - Relations and Functions, Maths, Class 12
Example Invertible functions-3 (Part - 34) - Relations and Functions, Maths, Class 12 Video Lecture
FAQs on Example Invertible functions-3 (Part - 34) - Relations and Functions, Maths, Class 12 Video Lecture
| 1. What is an invertible function? |  |
| 2. How do you determine if a function is invertible? | |
Ans. To determine if a function is invertible, we need to check if it is both injective (one-to-one) and surjective (onto). A function is injective if each element in the range of the function corresponds to a unique element in the domain. A function is surjective if every element in the range has a corresponding element in the domain. If a function satisfies both conditions, it is invertible.
| 3. Can every function be inverted? | |
Ans. No, not every function can be inverted. For a function to be invertible, it must be both injective (one-to-one) and surjective (onto). If a function is not injective, it means that multiple elements in the domain map to the same element in the range, making it impossible to uniquely determine the inverse. If a function is not surjective, it means that there are elements in the range that do not have a corresponding element in the domain, making it impossible to define an inverse.
| 4. How do you find the inverse of a function? | |
Ans. To find the inverse of a function, follow these steps:
1. Replace the function notation f(x) with y.
2. Swap the x and y variables.
3. Solve the resulting equation for y.
4. Replace y with the inverse function notation f^(-1)(x).
For example, if the original function is y = 2x + 3, we can find its inverse as follows:
1. Replace y with x: x = 2y + 3.
2. Swap x and y: y = 2x + 3.
3. Solve for y: y = (x - 3) / 2.
4. Replace y with the inverse function notation: f^(-1)(x) = (x - 3) / 2.
| 5. What is the significance of invertible functions? | |
Ans. Invertible functions have several significant properties. Firstly, the inverse of an invertible function undoes the original function, meaning that applying the original function and then its inverse will result in the original input. Secondly, invertible functions allow us to solve equations involving the function by applying its inverse. Furthermore, invertible functions provide a way to establish a correspondence between the domain and range, allowing for various mathematical and practical applications such as data encryption, signal processing, and optimization problems.
Text Transcript from Video
hello friends this video on relations
and functions part 34 is brought to you
by exam fear calm no more fear from exam
before watching this video please make
sure that you have watched part 1 - part
33 we have to tell whether the function
is worse or not we have three functions
if does this G this and H it is people
telling it f is invertible or not so
let's do the first one for if the
question says 1 2 3 4 it goes to 10 1
was due and f of 1 is 10 is 10 3 is 10
first you see this function is not 1
over y because all these values are
pointing to me ever since not one on it
is not invertible let's take G G is 5 6
7 8 goes to 1 2 3 4 5 goes to 4 6 goes
to 3 7 goes again to 4 in aid goes to
heroes that you see one for they are two
pointers coming also one has three
orphan so it is not Ando and it is not
even phenomenal so this function is also
not invertible let's say the last one H
1 H has value 2 3 4 5 & 7 9 11 goes to 7
3 goes to 9 who goes to limit by us do
you see this function this is one of 1
why because all these values have only
one pointer also it is on low Y so this
function is 1 or not low so it is
invertible this is one order it is by
ejected since it is by ejected this is
then going so what we will do if we are
to tell if the function is invertible or
not the first thing is ridiculously
we have to find where your function is
by JPEG or not then the function is 101
and on tour if the function is
one-to-one and onto that means the
function is so this is invertible and
this is why we have to show that the
function from minus 1 to 1 this is minus
1 to 1 this whole range R is any real
number any real number let's say is
given by FX is equal to X by X plus 2 so
here the value of x can be minus 1 and 1
they put minus 1 as well so this becomes
minus 1 by minus 1 plus 2 that is minus
1 so this minus 1 function is if I tell
1 because they're two possible values
plus 1 by 1 plus 2 that is 1 by 3 1
points you know that and there are so
many other number also like 1 by 2 like
for fish for which there is no memory
there are orphaned elements in this set
and this is my function f now find the
inverse of the function but please note
the inverse of the function is not in
this function the function inverse is
minus 1 to 1 do range of it this is our
so I am talking about this function
gives minus does minus 1 plus 1 and here
it is the range of F so what is the
range of F range of F is minus 1 and 1
by 3 so this is my function and I have
to find the inverse of this function
please note we are not asked to find the
inverse of this function f 2 1 we are
fine we are asked to find the inverse of
this function from minus 1 1 to range of
range of F is
this one bus so yes of course you find
inverse of this function fine it was the
first thing is whether it is 101 or on
tour or not so if you see this this
function is 101 and on why it is 101
because -1 has only one element pointing
to me and thus number three also has
only one element pointing here also on
to because there is no often limited but
if we notice this is not on to this
function is normal but this one is
wonder and this is the one we are asked
so that means it is invertible
therefore invertible correct so to find
the inverse just clip it so this becomes
minus 1/2 plus essentially and this
becomes minus 1 to plus 1 minus 1 goes
to 1 and plus 1 by 3 goes to plus this
is my inverse focus and that is what I
was supposed to find so we want to write
in this fashion and say something F
inverse is nothing but range of F 2
minus 1 plus 1 correct
this is my inverse function so I guess
this is G function G is equal to a
inverse inverse surgery or scheme
actually here in the same and this value
is nothing but G of Y is here Y is equal
to X Y this is all this comes out to be
Y by 1 minus y where Y is not equal to
obviously this becomes e bar that's up
on that is right and we have this
function from R to R given by FX equal
to 4x plus 3 we have to prove it is
invertible and also way to find out
inverse often they are to approve to
this question the first s became clue
that it is 101 Nando and then the
interval is invertible but that approach
is not
because anyway we have to find the
inverse what we can do you can take a
function G then we can find geo F and F
of G then we can prove it is invertible
so what we can do
the question says y is equal to where
it's firstly this one let's say X equal
a 1 y -3 so we can say that the let's
assume is a function G if also from R to
R you can say that G of Y is nothing but
y -1 but this is my function I need as
you mean this is the inverse of F now to
prove let's find G or F this is nothing
but G of F X this comes out to be G of X
plus 3 that comes out to be G of s 3 is
1 and F is nothing but G of Y is y minus
3 by 2 because spirity of G of Y this
comes out to e gy
this is f of G Y is nothing but Y minus
G by flow and there is nothing but f of
X that is y so we see I am getting G of
X is X and F of Y is y so that I can say
that F inverse what is e G is this
function this is the inverse of a
please clear at ancient health difficult
version there is a function X to Y and
the invertible function we have to show
that it has unique inverse who that it
is linear inverse let's assume it has 2
inverse let us use G 1 and G 2 V 2
inverse of and then we'll through that
our assumption was in kind all right so
if G 1 and G 2 are 2 inverse can see
that if G 1 is equal to F of G 2 and
that should be equal to I by cutting now
if F of G 1 is what F of G 2 I can say
that F of G 1 by Q 1 now since F is
invertible f is invertible since the
episode vertical f is 1 1 correct since
F is 101 we know that if f of X 1 is
equal to f of X 2 and F is 101 that
implies X 1 is go to extreme same thing
here instead of X 1 it is G 1 by here
instead of X 2 we have G 2 by L so from
this we can say that G 1 y is equal to
GT 1 then this part no issue since was F
was 1 over we could say that
Zeke advisable to divide this in this
form now since G 1 is also one of our
because G is nothing but inverse of this
function correct so G also inverse of
this
and Y is equal to Y by default you see
thus we can see that G 1 is equal to G
that is what the functions are same so
we assume that these g1 and g2 are two
different inverse of F but we found that
both are same function that this
invertible function will have a unique
inverse all right let's take for example
NR so there's a function from 1 2 3 to
ABC f1 is going to have musical given
and have please when this is here now we
have to find F inverse find if F inverse
the first thing we have to find whether
it is invertible or not so if you see
this is 101 because ABC all only one
finds are coming right so it is 1 1 also
it is on don't buy because there is no
extra element here there is no orphan
element in the coat of it or it is side
so it is invertible in word then well
they'll just flip this this is a fine F
inverse and slip it this becomes PVC and
here I get 1 2 3 a we point to 1 leaving
my boots in lights this is my F inverse
function
dandy now the question is we have to
prove that F inverse of F inverse equal
to M so let's find F inverse of this
guide is flip it once again flip this
guy once again this becomes 1 2 3 and
this becomes easy this is wrong
populations if you see this is nothing
but if F and s are exactly so we can see
that F inverse of F inverse is f self
thank you
visit exam viacom to watch free
education videos try free on my deaths
can the best quality study materials
study from the best tutors and mentors
and much more
thanks and screen
and functions part 34 is brought to you
by exam fear calm no more fear from exam
before watching this video please make
sure that you have watched part 1 - part
33 we have to tell whether the function
is worse or not we have three functions
if does this G this and H it is people
telling it f is invertible or not so
let's do the first one for if the
question says 1 2 3 4 it goes to 10 1
was due and f of 1 is 10 is 10 3 is 10
first you see this function is not 1
over y because all these values are
pointing to me ever since not one on it
is not invertible let's take G G is 5 6
7 8 goes to 1 2 3 4 5 goes to 4 6 goes
to 3 7 goes again to 4 in aid goes to
heroes that you see one for they are two
pointers coming also one has three
orphan so it is not Ando and it is not
even phenomenal so this function is also
not invertible let's say the last one H
1 H has value 2 3 4 5 & 7 9 11 goes to 7
3 goes to 9 who goes to limit by us do
you see this function this is one of 1
why because all these values have only
one pointer also it is on low Y so this
function is 1 or not low so it is
invertible this is one order it is by
ejected since it is by ejected this is
then going so what we will do if we are
to tell if the function is invertible or
not the first thing is ridiculously
we have to find where your function is
by JPEG or not then the function is 101
and on tour if the function is
one-to-one and onto that means the
function is so this is invertible and
this is why we have to show that the
function from minus 1 to 1 this is minus
1 to 1 this whole range R is any real
number any real number let's say is
given by FX is equal to X by X plus 2 so
here the value of x can be minus 1 and 1
they put minus 1 as well so this becomes
minus 1 by minus 1 plus 2 that is minus
1 so this minus 1 function is if I tell
1 because they're two possible values
plus 1 by 1 plus 2 that is 1 by 3 1
points you know that and there are so
many other number also like 1 by 2 like
for fish for which there is no memory
there are orphaned elements in this set
and this is my function f now find the
inverse of the function but please note
the inverse of the function is not in
this function the function inverse is
minus 1 to 1 do range of it this is our
so I am talking about this function
gives minus does minus 1 plus 1 and here
it is the range of F so what is the
range of F range of F is minus 1 and 1
by 3 so this is my function and I have
to find the inverse of this function
please note we are not asked to find the
inverse of this function f 2 1 we are
fine we are asked to find the inverse of
this function from minus 1 1 to range of
range of F is
this one bus so yes of course you find
inverse of this function fine it was the
first thing is whether it is 101 or on
tour or not so if you see this this
function is 101 and on why it is 101
because -1 has only one element pointing
to me and thus number three also has
only one element pointing here also on
to because there is no often limited but
if we notice this is not on to this
function is normal but this one is
wonder and this is the one we are asked
so that means it is invertible
therefore invertible correct so to find
the inverse just clip it so this becomes
minus 1/2 plus essentially and this
becomes minus 1 to plus 1 minus 1 goes
to 1 and plus 1 by 3 goes to plus this
is my inverse focus and that is what I
was supposed to find so we want to write
in this fashion and say something F
inverse is nothing but range of F 2
minus 1 plus 1 correct
this is my inverse function so I guess
this is G function G is equal to a
inverse inverse surgery or scheme
actually here in the same and this value
is nothing but G of Y is here Y is equal
to X Y this is all this comes out to be
Y by 1 minus y where Y is not equal to
obviously this becomes e bar that's up
on that is right and we have this
function from R to R given by FX equal
to 4x plus 3 we have to prove it is
invertible and also way to find out
inverse often they are to approve to
this question the first s became clue
that it is 101 Nando and then the
interval is invertible but that approach
is not
because anyway we have to find the
inverse what we can do you can take a
function G then we can find geo F and F
of G then we can prove it is invertible
so what we can do
the question says y is equal to where
it's firstly this one let's say X equal
a 1 y -3 so we can say that the let's
assume is a function G if also from R to
R you can say that G of Y is nothing but
y -1 but this is my function I need as
you mean this is the inverse of F now to
prove let's find G or F this is nothing
but G of F X this comes out to be G of X
plus 3 that comes out to be G of s 3 is
1 and F is nothing but G of Y is y minus
3 by 2 because spirity of G of Y this
comes out to e gy
this is f of G Y is nothing but Y minus
G by flow and there is nothing but f of
X that is y so we see I am getting G of
X is X and F of Y is y so that I can say
that F inverse what is e G is this
function this is the inverse of a
please clear at ancient health difficult
version there is a function X to Y and
the invertible function we have to show
that it has unique inverse who that it
is linear inverse let's assume it has 2
inverse let us use G 1 and G 2 V 2
inverse of and then we'll through that
our assumption was in kind all right so
if G 1 and G 2 are 2 inverse can see
that if G 1 is equal to F of G 2 and
that should be equal to I by cutting now
if F of G 1 is what F of G 2 I can say
that F of G 1 by Q 1 now since F is
invertible f is invertible since the
episode vertical f is 1 1 correct since
F is 101 we know that if f of X 1 is
equal to f of X 2 and F is 101 that
implies X 1 is go to extreme same thing
here instead of X 1 it is G 1 by here
instead of X 2 we have G 2 by L so from
this we can say that G 1 y is equal to
GT 1 then this part no issue since was F
was 1 over we could say that
Zeke advisable to divide this in this
form now since G 1 is also one of our
because G is nothing but inverse of this
function correct so G also inverse of
this
and Y is equal to Y by default you see
thus we can see that G 1 is equal to G
that is what the functions are same so
we assume that these g1 and g2 are two
different inverse of F but we found that
both are same function that this
invertible function will have a unique
inverse all right let's take for example
NR so there's a function from 1 2 3 to
ABC f1 is going to have musical given
and have please when this is here now we
have to find F inverse find if F inverse
the first thing we have to find whether
it is invertible or not so if you see
this is 101 because ABC all only one
finds are coming right so it is 1 1 also
it is on don't buy because there is no
extra element here there is no orphan
element in the coat of it or it is side
so it is invertible in word then well
they'll just flip this this is a fine F
inverse and slip it this becomes PVC and
here I get 1 2 3 a we point to 1 leaving
my boots in lights this is my F inverse
function
dandy now the question is we have to
prove that F inverse of F inverse equal
to M so let's find F inverse of this
guide is flip it once again flip this
guy once again this becomes 1 2 3 and
this becomes easy this is wrong
populations if you see this is nothing
but if F and s are exactly so we can see
that F inverse of F inverse is f self
thank you
visit exam viacom to watch free
education videos try free on my deaths
can the best quality study materials
study from the best tutors and mentors
and much more
thanks and screen
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