मराठी

Let A = [-1, 1]. Then, Discuss Whether the Following Function From A To Itself is One-one, Onto Or Bijective : G(X) = |X| - Mathematics

Advertisements
Advertisements

प्रश्न

Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : g(x) = |x|  

बेरीज
Advertisements

उत्तर

g(x) = |x|

Injection test:
Let x and y be any two elements in the domain (A), such that f(x) = f(y).

f(x) = f(y)

|x| = |y|

x = ± y

So, f is not one-one.

Surjection test :

For y = -1, there is no value of x in A.

So, f is not onto.

So, f is not bijective.

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 8.2 | पृष्ठ ३२

व्हिडिओ ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्‍न

Check the injectivity and surjectivity of the following function:

f : R → R given by f(x) = x2


Let A = {1, 2, 3}, B = {4, 5, 6, 7} and let f = {(1, 4), (2, 5), (3, 6)} be a function from A to B. Show that f is one-one.


Show that the function f : R → R given by f(x) = x3 is injective.


Give an example of a function which is neither one-one nor onto ?


Let A = {−1, 0, 1} and f = {(xx2) : x ∈ A}. Show that f : A → A is neither one-one nor onto.


Let A = [-1, 1]. Then, discuss whether the following function from A to itself is one-one, onto or bijective : `f (x) = x/2`


Set of ordered pair of a function ? If so, examine whether the mapping is injective or surjective :{(ab) : a is a person, b is an ancestor of a


Show that the logarithmic function  f : R0+ → R   given  by f (x)  loga x ,a> 0   is   a  bijection.


Given A = {2, 3, 4}, B = {2, 5, 6, 7}. Construct an example of each of the following:
(i) an injective map from A to B
(ii) a mapping from A to B which is not injective
(iii) a mapping from A to B.


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


Find gof and fog when f : R → R and g : R → R is defined by  f(x) = 2x + x2 and  g(x) = x3


Let R+ be the set of all non-negative real numbers. If f : R+ → R+ and g : R+ → R+ are defined as `f(x)=x^2` and `g(x)=+sqrtx` , find fog and gof. Are they equal functions ?


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


State with reason whether the following functions have inverse :

g : {5, 6, 7, 8} → {1, 2, 3, 4} with g = {(5, 4), (6, 3), (7, 4), (8, 2)}


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


Let A and B be two sets, each with a finite number of elements. Assume that there is an injective map from A to B and that there is an injective map from B to A. Prove that there is a bijection from A to B.


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


If f : A → Ag : A → A are two bijections, then prove that fog is a surjection ?


If f : R → R is given by f(x) = x3, write f−1 (1).


If f : C → C is defined by f(x) = x4, write f−1 (1).


If f : R → Rg : R → are given by f(x) = (x + 1)2 and g(x) = x2 + 1, then write the value of fog (−3).


If f : R → R is defined by f(x) = 3x + 2, find f (f (x)).


The range of the function

\[f\left( x \right) =^{7 - x} P_{x - 3}\]

 


If  \[f : R \to \left( - 1, 1 \right)\] is defined by

\[f\left( x \right) = \frac{- x|x|}{1 + x^2}, \text{ then } f^{- 1} \left( x \right)\] equals

 


If \[f : R \to R\] is given by \[f\left( x \right) = x^3 + 3, \text{then} f^{- 1} \left( x \right)\] is equal to

 


Mark the correct alternative in the following question:
Let A = {1, 2, ... , n} and B = {a, b}. Then the number of subjections from A into B is


Mark the correct alternative in the following question:
If the set A contains 7 elements and the set B contains 10 elements, then the number one-one functions from A to B is


If A = {a, b, c, d} and f = {a, b), (b, d), (c, a), (d, c)}, show that f is one-one from A onto A. Find f–1


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


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

h(x) = x|x|


Which of the following functions from Z into Z is bijective?


Let f : R → R be a function defined by f(x) `= ("e"^abs"x" - "e"^-"x")/("e"^"x" + "e"^-"x")` then f(x) is


A general election of Lok Sabha is a gigantic exercise. About 911 million people were eligible to vote and voter turnout was about 67%, the highest ever


Let I be the set of all citizens of India who were eligible to exercise their voting right in the general election held in 2019. A relation ‘R’ is defined on I as follows:

R = {(V1, V2) ∶ V1, V2 ∈ I and both use their voting right in the general election - 2019}

  • Three friends F1, F2, and F3 exercised their voting right in general election-2019, then which of the following is true?

Function f: R → R, defined by f(x) = `x/(x^2 + 1)` ∀ x ∈ R is not


If f; R → R f(x) = 10x + 3 then f–1(x) is:


The solution set of the inequation log1/3(x2 + x + 1) + 1 > 0 is ______.


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.


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.


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×