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Test: Introduction To Functions - JEE MCQ


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15 Questions MCQ Test Mathematics (Maths) for JEE Main & Advanced - Test: Introduction To Functions

Test: Introduction To Functions for JEE 2024 is part of Mathematics (Maths) for JEE Main & Advanced preparation. The Test: Introduction To Functions questions and answers have been prepared according to the JEE exam syllabus.The Test: Introduction To Functions MCQs are made for JEE 2024 Exam. Find important definitions, questions, notes, meanings, examples, exercises, MCQs and online tests for Test: Introduction To Functions below.
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Test: Introduction To Functions - Question 1

If y is expressed in terms of a variable x as Y = ƒ(x), then y is called

Detailed Solution for Test: Introduction To Functions - Question 1

If y is expressed in terms of a variable x as y=f(x), then y is called explicit function.

Test: Introduction To Functions - Question 2

Let f : R → R defined as f(x) = x be an identity function. Then

Detailed Solution for Test: Introduction To Functions - Question 2

Here, we are given a function f ,which is defined from R to R. Also, f is said to be the Identity Function.
The Identity Function returns the same output as given in the input. 
That is, if you input some x into f(x), you get the result as x itself. So, the Identity Function is:

As we can see, the Domain of the function is R , and the co-domain as well as range is also R.

Test: Introduction To Functions - Question 3

Let A = {1, 2, 3} and B = {5, 6, 7, 8, 9} and let f(x) = {(1, 8), (2, 7), (3, 6)} then f is

Detailed Solution for Test: Introduction To Functions - Question 3

Here, f(1) = 8, f(2) = 7, f(3) = 6.

Since, different points of domain have the different f-image in the range, therefore f is a one-one function.

Test: Introduction To Functions - Question 4

In the mapping above, the function f4 is an:

Detailed Solution for Test: Introduction To Functions - Question 4

Clearly, mappings given in options (a),(b) and (c ) satisfy the given conditions and are one-one onto.

Test: Introduction To Functions - Question 5

Let f : N →N be defined by   for all n ∈ N.
Then, the function f is

Detailed Solution for Test: Introduction To Functions - Question 5

F(n) = n+2 if n is odd
so if n=1, then f(1)= 3
if n=3 , f(3)=5
On the other hand
F(n)= n+3 if n is even
if n= 2,f(2)=5
if n=4 ,f(4)= 7
Clearly f(3) = f(2) = 5
but 3 does not equals to 3, so it is not one one i.e. injective also 1€N
 i.e. codomain
but here 1€ range does not equals to codomain so it is not surjective.

Test: Introduction To Functions - Question 6

A function f: R → R defined as f(x) = x4 is​

Detailed Solution for Test: Introduction To Functions - Question 6

 f: R → R is defined as f(x) = x4
Let x, y ∈ R such that f(x) = f(y).
=> x4 = y4
=> x =+-y
Therefore, f(x1) = f(x2) does not imply that x1=x2.
For instance,f(1) = f(-1) = 1
∴ f is not one-one.
Consider an element 2 in co-domain R. It is clear that there does not exist any x in domain R such that f(x) = 2.
∴ f is not onto.
Hence, function f is neither one-one nor onto.

Test: Introduction To Functions - Question 7

A function f: X → Y is bijective if and only if​

Detailed Solution for Test: Introduction To Functions - Question 7

A bijective function is also known as an invertible function. It is a function that has both one-to-one and onto properties.

Test: Introduction To Functions - Question 8

A function f: X → Y is injective if and only if

Detailed Solution for Test: Introduction To Functions - Question 8

Let X, Y be sets, and let f : X → Y be a function. We say that f is injective (sometimes called one-to-one) if ∀x1, x2 ∈ X, f(x1) = f(x2
⇒ x1 = x2.

Test: Introduction To Functions - Question 9

A function f: R → R defined as f(x) = 5x is​

Detailed Solution for Test: Introduction To Functions - Question 9

Suppose x1 and x2 are real numbers such that f(x1) = f(x2). (We need to show x1 = x2 .)
5x1  = 5x2 
Dividing by 5 on both sides gives
x1 = x2  (function is one - one)
Let y ∈R. (We need to show that  x in R such that f(x) = y.)
If such a real number x exists, then 5x -2 = y and x = (y + 2)/5. x is a real number since sums and quotients (except for division by 0) of real numbers are real numbers. It follows that
f(x) = 5((y + 2)/5)          
by the substitution and the definition of f
       = y + 2 
       = y                
by basic algebra
Hence, f is onto.

Test: Introduction To Functions - Question 10

A function f: X → Y is surjective if and only if​

Detailed Solution for Test: Introduction To Functions - Question 10

A function f (from set X to Y) is surjective if and only if for every y in Y, there is at least one x in X such that f(x) = y, in other words f is surjective if and only if f(X) = Y.

Test: Introduction To Functions - Question 11

The range of function f : R → R defined by f(x) = x2 is​

Detailed Solution for Test: Introduction To Functions - Question 11

As x is square of every real number, so the f(x) = x2 cannot be the negative.

Test: Introduction To Functions - Question 12

Let f : R → R defined as f(x) = 5 be a constant function. Then​

Detailed Solution for Test: Introduction To Functions - Question 12

It's because f(x) has a constant value, and range will always be 5.

Test: Introduction To Functions - Question 13

The function, f(x) = 2x - 1 is

Detailed Solution for Test: Introduction To Functions - Question 13

The function f(x) = 2x - 1 is surjective.

Explanation:

A function is surjective (also called onto) if for every element y in the codomain, there exists an element x in the domain such that f(x) = y.

In this case, the function f(x) = 2x - 1 is a linear function. For any given y value, we can find an x value such that f(x) = y:

y = 2x - 1
x = (y + 1) / 2

So, for any y value, there exists an x value that satisfies the equation. Therefore, the function is surjective.

Test: Introduction To Functions - Question 14

Let f: {1, 2, 3} → {1, 2, 3} be an onto function. Then, f is​

Detailed Solution for Test: Introduction To Functions - Question 14

(i) f(1)=1,f(2)=2,f(3)=3
(ii) f(1)=1,f(2)=3,f(3)=2
(iii) f(1)=2,f(2)=3,f(3)=1
(iv) f(1)=2,f(2)=1,f(3)=3
(v) f(1)=3,f(2)=1,f(3)=2
(vi) f(1)=3,f(2)=2,f(3)=1
Since, f is onto, all elements of {1,2,3} have unique pre-image.
The above cases are possible.
Since, every element 1,2,3 has either of image 1,2,3 and that image is unique.
∴  f is one-one.
∴ Function f:A→A is one-one.

Test: Introduction To Functions - Question 15

A function f: Z → Z defined as f(x) = x3 is​

Detailed Solution for Test: Introduction To Functions - Question 15

Explanation:


 


  • For a function to be one-one, each element in the domain must map to a unique element in the co-domain.

  • For a function to be onto, each element in the co-domain must have at least one element in the domain that maps to it.

  • Given function f: Z → Z defined as f(x) = x3

  • Since f(x) = f(-x) for all x in Z, the function is not one-one.

  • However, for any y in Z, there exists an x such that f(x) = y. For example, f(2) = 8, f(-2) = -8, so the function is onto.

  • Therefore, the function f: Z → Z defined as f(x) = x^3 is one-one but not onto.


  •  
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