मराठी

Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.

Advertisements
Advertisements

प्रश्न

Let the function f: R → R be defined by f(x) = 4x – 1, ∀ x ∈ R. Then, show that f is one-one.

बेरीज
Advertisements

उत्तर

For any two elements x1 , x2 ∈ R such that f(x1) = f(x2),

We have 4x1 – 1 = 4x2 – 1

⇒ 4x1 = 4x2 

i.e., x1 = x2

Hence f is one-one.

shaalaa.com
  या प्रश्नात किंवा उत्तरात काही त्रुटी आहे का?
पाठ 1: Relations And Functions - Solved Examples [पृष्ठ ३]

APPEARS IN

एनसीईआरटी एक्झांप्लर Mathematics Exemplar [English] Class 12
पाठ 1 Relations And Functions
Solved Examples | Q 4 | पृष्ठ ३

व्हिडिओ ट्यूटोरियलVIEW ALL [5]

संबंधित प्रश्‍न

Let f : N → N be defined by f(n) = `{((n+1)/2", if n is odd"),(n/2", if n is even"):}` for all n ∈ N.

State whether the function f is bijective. Justify your answer.


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


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

(Hint: Consider f(x) = x and g(x) =|x|)


Let S = {a, b, c} and T = {1, 2, 3}. Find F−1 of the following functions F from S to T, if it exists.

F = {(a, 3), (b, 2), (c, 1)} 


Prove that the function f : N → N, defined by f(x) = x2 + x + 1, is one-one but not onto


Classify the following function as injection, surjection or bijection : f : N → N given by f(x) = x3


If f : R → R be the function defined by f(x) = 4x3 + 7, show that f is a bijection.


Let f = {(3, 1), (9, 3), (12, 4)} and g = {(1, 3), (3, 3) (4, 9) (5, 9)}. Show that gof and fog are both defined. Also, find fog and gof.


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


Find fog and gof  if : f (x) = x+1, g (x) = sin x .


Let f(x) = x2 + x + 1 and g(x) = sin x. Show that fog ≠ gof.


If f(x) = sin x and g(x) = 2x be two real functions, then describe gof and fog. Are these equal functions?


If f : R → (−1, 1) defined by `f (x) = (10^x- 10^-x)/(10^x + 10 ^-x)` is invertible, find f−1.


Let A = {1, 2, 3, 4} and B = {a, b} be two sets. Write the total number of onto functions from A to B.


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


Let M be the set of all 2 × 2 matrices with entries from the set R of real numbers. Then, the function f : M→ R defined by f(A) = |A| for every A ∈ M, 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


\[f : Z \to Z\]  be given by

 ` f (x) = {(x/2, ", if  x is even" ) ,(0 , ", if  x  is  odd "):}`

Then,  f is


The inverse of the function

\[f : R \to \left\{ x \in R : x < 1 \right\}\] given by

\[f\left( x \right) = \frac{e^x - e^{- x}}{e^x + e^{- x}}\] is 

 


Let

 \[A = \left\{ x \in R : x \geq 1 \right\}\] The inverse of the function, 

\[f : A \to A\] given by

\[f\left( x \right) = 2^{x \left( x - 1 \right)} , is\]

 


If the function

\[f : R \to R\]  be such that

\[f\left( x \right) = x - \left[ x \right]\] where [x] denotes the greatest integer less than or equal to x, then \[f^{- 1} \left( x \right)\]

 


Let  \[f\left( x \right) = \frac{\alpha x}{x + 1}, x \neq - 1\] Then, for what value of α is \[f \left( f\left( x \right) \right) = x?\]

 


Set A has 3 elements and the set B has 4 elements. Then the number of injective mappings that can be defined from A to B is ______.


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


Let f: R → R be the function defined by f(x) = 2x – 3 ∀ x ∈ R. write f–1 


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)}


The function f: R → R defined as f(x) = x3 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}

  • Mr. ’X’ and his wife ‘W’ both exercised their voting right in the general election-2019, Which of the following is true?

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 ____________.

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


If f: [0, 1]→[0, 1] is defined by f(x) = `(x + 1)/4` and `d/(dx) underbrace(((fofof......of)(x)))_("n"  "times")""|_(x = 1/2) = 1/"m"^"n"`, m ∈ N, then the value of 'm' is ______.


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


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


In the function from \(X\) to \(Y\), if \(6\in Y\) is not the image of any element of \(X\), the function is:


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


A perfect one-to-one matching between users and unique IDs is similar to:


Which condition represents an onto (surjective) function?


Which condition represents an into function?


Share
Notifications

Englishहिंदीमराठी


      Forgot password?
Use app×