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Invertible Functions Video Lecture | Mathematics (Maths) Class 12 - JEE
FAQs on Invertible Functions Video Lecture - Mathematics (Maths) Class 12 - JEE
| 1. What is an invertible function? |  |
Ans. An invertible function, also known as a one-to-one function, is a function that maps each element of its domain to a unique element in its range. In other words, for every input, there is exactly one output, and vice versa.
| 2. How can you determine if a function is invertible? | |
Ans. To determine if a function is invertible, you can use the horizontal line test. If no horizontal line intersects the graph of the function more than once, then the function is invertible. Additionally, a function must also be one-to-one, meaning that it passes the vertical line test, where no vertical line intersects the graph more than once.
| 3. What is the inverse of an invertible function? | |
Ans. The inverse of an invertible function is a new function that undoes the actions of the original function. In other words, if the original function maps an input x to an output y, then the inverse function maps y back to x. The inverse function is denoted as f^(-1)(x).
| 4. Can all functions be inverted? | |
Ans. No, not all functions can be inverted. Only functions that are one-to-one, or injective, can be inverted. If a function is not one-to-one, then there would be multiple inputs that map to the same output, making it impossible to determine the original input from the output.
| 5. How can you find the inverse of an invertible function? | |
Ans. To find the inverse of an invertible function, you can follow these steps:
1. Replace the function's notation with y.
2. Swap the roles of x and y in the equation.
3. Solve the equation for y to express it in terms of x.
4. Replace y with f^(-1)(x) to denote the inverse function.
5. Verify that the inverse function is indeed the inverse by using composition, where applying the original function and its inverse to an input should yield the original input.
Text Transcript from Video
convertible function let F be a function
from one to three to ABC and f is one
one and on to such that F of one is
equal to a F of two is equal to b f of 3
is equal to C there exists a function G
from ABC 2 1 2 3 touch that G of a is
equal to 1 G of B is equal to 2 G of C
is equal to 3 now let us examine G Oh F
and f or g between such functions let us
find geo F of 1 G of F 1 F of 1 is e and
g o f of a is equal to 1 so G or F of 1
is 1 similarly G oh F of 2 is 2 and G Oh
F of 3 is 3 it means do FX is equal to X
which is an identity function on X now
let us find f OG of a F or G of a is
equal to F of G a G of a is equal to 1
and F of 1 is equal to a similarly F or
G of B is B and F Oh G of C is C it
means f or g y is equal to y or it is an
identity function on Y it means function
f from X to Y is invertible if there
exists a function G from Y to X such
that G Oh F is equal to I X and F or G
is equal to I why function G is called
the inverse of function f and it is
denoted by F raised to the power minus 1
for a function to be invertible it must
be one one and onto or we can say if f
is one one and onto then f must be
invertible
from one to three to ABC and f is one
one and on to such that F of one is
equal to a F of two is equal to b f of 3
is equal to C there exists a function G
from ABC 2 1 2 3 touch that G of a is
equal to 1 G of B is equal to 2 G of C
is equal to 3 now let us examine G Oh F
and f or g between such functions let us
find geo F of 1 G of F 1 F of 1 is e and
g o f of a is equal to 1 so G or F of 1
is 1 similarly G oh F of 2 is 2 and G Oh
F of 3 is 3 it means do FX is equal to X
which is an identity function on X now
let us find f OG of a F or G of a is
equal to F of G a G of a is equal to 1
and F of 1 is equal to a similarly F or
G of B is B and F Oh G of C is C it
means f or g y is equal to y or it is an
identity function on Y it means function
f from X to Y is invertible if there
exists a function G from Y to X such
that G Oh F is equal to I X and F or G
is equal to I why function G is called
the inverse of function f and it is
denoted by F raised to the power minus 1
for a function to be invertible it must
be one one and onto or we can say if f
is one one and onto then f must be
invertible
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