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Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______

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प्रश्न

Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is ______

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उत्तर

Consider the set A containing n elements. Then, the total number of injective functions from A onto itself is n!

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अध्याय 1: Relations And Functions - Solved Examples [पृष्ठ १०]

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एनसीईआरटी एक्झांप्लर Mathematics Exemplar [English] Class 12
अध्याय 1 Relations And Functions
Solved Examples | Q 27 | पृष्ठ १०

वीडियो ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्न

Show that the function f in `A=R-{2/3} ` defined as `f(x)=(4x+3)/(6x-4)` is one-one and onto hence find f-1


Let A = R – {3} and B = R – {1}. Consider the function f : A → B defined by f(x) = `((x - 2)/(x - 3))`. Is f one-one and onto? Justify your answer.


Let f: R → R be defined as f(x) = 10x + 7. Find the function g: R → R such that g o f = f o g = 1R.


Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = 3 − 4x


Find the number of all onto functions from the set A = {1, 2, 3, ..., n} to itself.


Suppose f1 and f2 are non-zero one-one functions from R to R. Is `f_1 / f^2` necessarily one - one? Justify your answer. Here,`f_1/f_2 : R → R   is   given   by   (f_1/f_2) (x) = (f_1(x))/(f_2 (x))  for all  x in R .`


Find gof and fog when f : R → R and g : R → R is defined by f(x) = 2x + 3 and  g(x) = x2 + 5 .


Let A = {a, b, c}, B = {u v, w} and let f and g be two functions from A to B and from B to A, respectively, defined as :
f = {(a, v), (b, u), (c, w)}, g = {(u, b), (v, a), (w, c)}.
Show that f and g both are bijections and find fog and gof.


Find fog and gof  if : f(x) = c, c ∈ R, g(x) = sin `x^2`


if f (x) = `sqrt (x +3) and  g (x) = x ^2 + 1` be two real functions, then find fog and gof.


Consider f : R → R given by f(x) = 4x + 3. Show that f is invertible. Find the inverse of f.


If f : A → A, g : A → A are two bijections, then prove that fog is an injection ?


Let A = {x ∈ R : −4 ≤ x ≤ 4 and x ≠ 0} and f : A → R be defined by \[f\left( x \right) = \frac{\left| x \right|}{x}\]Write the range of f.


Write the domain of the real function

`f (x) = 1/(sqrt([x] - x)`.


If f(x) = 4 −( x - 7)3 then write f-1 (x).


The function f : R → R defined by

`f (x) = 2^x + 2^(|x|)` is 

 


The function

\[f : R \to R\] defined by\[f\left( x \right) = \left( x - 1 \right) \left( x - 2 \right) \left( x - 3 \right)\]

(a) one-one but not onto
(b) onto but not one-one
(c) both one and onto
(d) neither one-one nor onto


The function

\[f : R \to R, f\left( x \right) = x^2\]
 

If  \[F : [1, \infty ) \to [2, \infty )\] is given by

\[f\left( x \right) = x + \frac{1}{x}, then f^{- 1} \left( x \right)\]

 


Let

\[f : [2, \infty ) \to X\] be defined by

\[f\left( x \right) = 4x - x^2\] Then, f is invertible if X =

 


Mark the correct alternative in the following question:
Let f : R  \[-\] \[\left\{ \frac{3}{5} \right\}\] \[\to\]  R be defined by f(x) = \[\frac{3x + 2}{5x - 3}\] Then,

 


If f(x) = `(x+3)/(4x−5) , "g"(x) = (3+5x)/(4x−1)` then verify that `("fog") (x)` = x.


Let X = {1, 2, 3}and Y = {4, 5}. Find whether the following subset of X ×Y are function from X to Y or not

f = {(1, 4), (1, 5), (2, 4), (3, 5)}


Let A = R – {3}, B = R – {1}. Let f: A → B be defined by f(x) = `(x - 2)/(x - 3)` ∀ x ∈ A . Then show that f is bijective.


Let f: R → R be defined by f(x) = `1/x` ∀ x ∈ R. Then f is ______.


If f(x) = (4 – (x – 7)3}, then f–1(x) = ______.


The smallest integer function f(x) = [x] is ____________.


If N be the set of all-natural numbers, consider f: N → N such that f(x) = 2x, ∀ x ∈ N, then f is ____________.


Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.

Answer the following questions using the above information.

  • Let f: R → R be defined by f(x) = x2 is:

Raji visited the Exhibition along with her family. The Exhibition had a huge swing, which attracted many children. Raji found that the swing traced the path of a Parabola as given by y = x2.

Answer the following questions using the above information.

  • Let f: {1,2,3,....} → {1,4,9,....} be defined by f(x) = x2 is ____________.

A function f: x → y is said to be one – one (or injective) if:


Difference between the greatest and least value of f(x) = `(1 + (cos^-1x)/π)^2 - (1 + (sin^-1x)/π)^2` is ______.


The graph of the function y = f(x) is symmetrical about the line x = 2, then ______.


The function f(x) = [x], where [x] denotes the greatest integer less than or equal to x; is continuous at ______.


Let f(x) be a polynomial function of degree 6 such that `d/dx (f(x))` = (x – 1)3 (x – 3)2, then

Assertion (A): f(x) has a minimum at x = 1.

Reason (R): When `d/dx (f(x)) < 0, ∀  x ∈ (a - h, a)` and `d/dx (f(x)) > 0, ∀  x ∈ (a, a + h)`; where 'h' is an infinitesimally small positive quantity, then f(x) has a minimum at x = a, provided f(x) is continuous at x = a.


Let \(X=\{-1,2,5,8,11\}\), \(Y=\{4,6,8\}\), and suppose different elements \(2,5,8\) of \(X\) have the same image \(4\). What type of function is it with respect to the one-one property?


If every seat in a classroom is occupied by one student, the situation resembles:


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