मराठी

Let A = {1, 2, 3}. Write All One-one From A To Itself. - Mathematics

Advertisements
Advertisements

प्रश्न

Let A = {1, 2, 3}. Write all one-one from A to itself.

बेरीज

उत्तर

A ={1, 2, 3}
Number of elements in  = 3
Number of one-one functions = number of ways of arranging 3 elements = 3! = 6

(i) {(1, 1), (2, 2), (3, 3)}
(ii) {(1, 1), (2, 3), (3, 2)}
(iii) {(1, 2 ), (2, 2), (3, 3 )}
(iv) {(1, 2), (2, 1), (3, 3)}
(v) {(1, 3), (2, 2), (3, 1)}
(vi) {(1, 3), (2, 1), (3,2 )}

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 2: Functions - Exercise 2.1 [पृष्ठ ३२]

APPEARS IN

आरडी शर्मा Mathematics [English] Class 12
पाठ 2 Functions
Exercise 2.1 | Q 10 | पृष्ठ ३२

व्हिडिओ ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्‍न

Prove that the greatest integer function f: → R, given by f(x) = [x], is neither one-one nor onto, where [x] denotes the greatest integer less than or equal to x.


Let f: R → R be defined as f(x) = 3x. Choose the correct answer.


If the function `f(x) = sqrt(2x - 3)` is invertible then find its inverse. Hence prove that `(fof^(-1))(x) = x`


Classify the following function as injection, surjection or bijection :

 f : R → R, defined by f(x) = x3 − x


Classify the following function as injection, surjection or bijection :

f : R → R, defined by f(x) = sin2x + cos2x


Let f : N → N be defined by

`f(n) = { (n+ 1, if n  is  odd),( n-1 , if n  is  even):}`

Show that f is a bijection. 

                      [CBSE 2012, NCERT]


Give examples of two functions f : N → Z and g : Z → Z, such that gof is injective but gis not injective.


If f : A → B and g : B → C are one-one functions, show that gof is a one-one function.


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


   if `f (x) = sqrt(1-x)` and g(x) = `log_e` x are two real functions, then describe functions fog and gof.


Let A = R - {3} and B = R - {1}. Consider the function f : A → B defined by f(x) = `(x-2)/(x-3).`Show that f is one-one and onto and hence find f-1.

                    [CBSE 2012, 2014]


Consider the function f : R→  [-9 , ∞ ]given by f(x) = 5x2 + 6x - 9. Prove that f is invertible with -1 (y) = `(sqrt(54 + 5y) -3)/5`             [CBSE 2015]


Let C denote the set of all complex numbers. A function f : C → C is defined by f(x) = x3. Write f−1(1).


If f : C → C is defined by f(x) = (x − 2)3, write f−1 (−1).


Let f be an invertible real function. Write ( f-1  of ) (1) + ( f-1  of ) (2) +..... +( f-1 of ) (100 )


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


Let

\[f : R - \left\{ n \right\} \to R\]

\[f\left( x \right) = \frac{x - m}{x - n}, \text{where} \ m \neq n .\] Then,
 

Let f, g: R → R be two functions defined as f(x) = |x| + x and g(x) = x – x ∀ x ∈ R. Then, find f o g and g o f


Let A be a finite set. Then, each injective function from A into itself is not surjective.


For sets A, B and C, let f: A → B, g: B → C be functions such that g o f is injective. Then both f and g are injective functions.


Are the following set of ordered pairs functions? If so, examine whether the mapping is injective or surjective.
{(a, b): a is a person, b is an ancestor of a}


Let the function f: R → R be defined by f(x) = cosx, ∀ x ∈ R. Show that f is neither one-one nor onto


Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

h(x) = x|x|


Let A = [–1, 1]. Then, discuss whether the following functions defined on A are one-one, onto or bijective:

k(x) = x2 


Let f: R → R be the functions defined by f(x) = x3 + 5. Then f–1(x) is ______.


Let A = {0, 1} and N be the set of natural numbers. Then the mapping f: N → A defined by f(2n – 1) = 0, f(2n) = 1, ∀ n ∈ N, is onto.


Let f : R → R be defind by f(x) = `1/"x"  AA  "x" in "R".` Then f is ____________.


Let f : R `->` R be a function defined by f(x) = x3 + 4, then f is ______.


An organization conducted a bike race under 2 different categories-boys and girls. Totally there were 250 participants. Among all of them finally, three from Category 1 and two from Category 2 were selected for the final race. Ravi forms two sets B and G with these participants for his college project. Let B = {b1,b2,b3} G={g1,g2} where B represents the set of boys selected and G the set of girls who were selected for the final race.

Ravi decides to explore these sets for various types of relations and functions.

  • Let R: B → G be defined by R = { (b1,g1), (b2,g2),(b3,g1)}, then R 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: N → N 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.

  • The function f: Z → Z defined by f(x) = x2 is ____________.

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


Let [x] denote the greatest integer ≤ x, where x ∈ R. If the domain of the real valued function f(x) = `sqrt((|[x]| - 2)/(|[x]| - 3)` is (–∞, a) ∪ [b, c) ∪ [4, ∞), a < b < c, then the value of a + b + c is ______.


For x ∈ R, x ≠ 0, let f0(x) = `1/(1 - x)` and fn+1 (x) = f0(fn(x)), n = 0, 1, 2, .... Then the value of `f_100(3) + f_1(2/3) + f_2(3/2)` is equal to ______.


Write the domain and range (principle value branch) of the following functions:

f(x) = tan–1 x.


Which one of the following graphs is a function of x?

Graph A Graph B

If f : R `rightarrow` R is defined by `f(x) = (2x - 7)/4`, show that f(x) is one-one and onto.


The trigonometric equation tan–1x = 3tan–1 a has solution for ______.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×