JEE Main Functions in One Shot - Amazon Web Services

JEE Main Functions in One Shot

JEE MAIN APRIL 2020 Super Revision PLAN

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Chapter name 7th Jan-I

7th Jan-II

8th Jan-I

8th Jan-II

9th Jan-I

9th Jan-II

Sets, Relation & Functions 1 0 1 1 1 0

JEE MAIN 2020

Chapter name

9th jan(I)

9th jan(II)

10th jan(I)

10th jan(II)

11th jan(I)

11th jan(II)

12th jan(I)

12th jan(II)

Sets, Relation & Functions

2 2 0 2 1 1 2 1

JEE MAIN 2019

JEE MAIN PAST YEARS CHAPTERWISE WEIGHTAGE

JEE MAIN 2019

Chapter name

8th Apr(I)

8th Apr(II)

9th April(I)

9th April(II)

10th April(I)

10th April(II)

12th Apr(I)

12th April(II)

Sets, Relation & Functions

1 2 0 2 2 1 1 0

Chapter name 2018 2017 2016 2015

Sets, Relation & Functions 1 2 2 0

JEE MAIN PAST YEARS CHAPTERWISE WEIGHTAGE

Definition of a FunctionLet A and B be two non empty sets.A function from A to B i.e. f : A ⟶ B is a relation such that(a) all the elements of A are related to the elements of B and(b) no element of A is related to more than one element of B

Both are relations of course from A to B.

A B

f

A B

f

Let’s play a FUNCTIONS Game !!!





Rules of finding domain1

expression , then expression ≠ 01.

Expression√ , then Expression ≥ 02.

f(x)4. Domain is D1

g(x) Domain is D2

Domain is D1 ∩ D2f(x) ± g(x)

loga x3. , then x > 0 and a > 0 and a ≠ 1

DOMAIN OF A FUNCTION

RANGE OF A FUNCTION

y = f (x)

x = g (y)

Now, values of y for which g (y) i.e. x is defined is the required range

Rules of finding range

Say we need to find range of y = f (x)

Try to express it in following form

Consider

EVEN – ODD FUNCTION

A function f(x) is said to be even function; if f(– x) = f(x) ∀ x

Graph of an even function is symmetric about y – axis

A function f(x) is said to odd; if f(– x) = – f(x) ∀ x.

ONE –ONE AND MANY ONE FUNCTIONS

A function f:A → B is said to be one-one or injectiveif all elements of A have different images in B

Otherwise it is called many one function.

Mathematicallyif f (a) = f (b) ⇒ a = bthen f (x) is 1-1 (or injective)

● A function f(x) is said to be periodic function, if there exists a positive real number such that .

● Least such value of ‘T’ is called Fundamental Period of y = f(x).● Graph of periodic function repeats at fixed length of interval.

Methods to find Period

(1) If period of f(x) is T then period of y = kf (ax + b) + c is

(2) If period of f(x) is T1 and period of g(x) is T2 then LCM (T1, T2) is period

of f(x) + g(x). [It need not be fundamental period].

Periodic Function

● A function f(x) is said to be periodic function, if there exists a positive real number such that .

● Least such value of ‘T’ is called Fundamental Period of y = f(x).● Graph of periodic function repeats at fixed length of interval.

MODULUS FUNCTION

–a a

a

y = x

y = –x

Properties

1) a ≤ | a |

2) | ab | = | a | | b |

3) | a + b | ≤ | a | + | b |

If a > 0; then equality holds

4) | a – b | ≥ | || a | – | b |3 and 4 are called triangle inequalities

In 3 and 4, equality holds if ab ≥ 0

GREATEST INTEGER FUNCTION 321

0–3 –2 –1 1 2 3

–2–1

–3

Domain = Set of all real numbers

Range = Set of integers

Properties

1) [x] + [–x] =0 ; x ∈ Z

–1 ; x ∉ Z

2) [x + k] [x] + k, for k∈ Z =

[kx] ≠ k[x]

Graph

–2

–1

0 1 2 3

1

Domain : Set of all real numbers

Range : [ 0, 1 )

x + 2

x + 1 x x – 1

x – 2 { x } = x – [ x ]

x + 1 ; –1≤ x < 0

x ; 0≤ x < 1x – 1 ; 1≤ x < 2

Fractional Part Function

EXPONENTIAL FUNCTION

y = ax

a > 1 0 < a < 1

Increasing function Decreasing function

a > 0 ; a ≠ 1

If x is increasing Then y is increasing

If x is increasing Then y is decreasing

LOGARITHMIC FUNCTION

y = logax

log a b = c if b = ac

log28 = 3

log5625 = 4

(23 = 8)

(54= 625)

y = logax ; a > 0 ; a ≠ 1

1

1

Increasing function

Decreasingfunction

a > 1

0 < a < 1

Both have range: all real

Onto and into functions

A function f(x) from A to B is said to be onto or surjectivefunction if every element of B is image of some element of A.

i.e. Range of f(x) = Co – domain of f(x)

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Q1. If g(x) = x2 + x -1 and (gof)(x) = 4x2 - 10x + 5, then is equal to :

7th Jan 2020- Shift I

A

B

D

C

JEE MAIN Functions: Super JEE Revision

Solution:

Q1. If g(x) = x2 + x -1 and (gof)(x) = 4x2 - 10x + 5, then is equal to :

7th Jan 2020- Shift I

A

B

D

C


Q2. The inverse function of , x (-1, 1) , is

7th Jan 2020- Shift II

A

B

D

C


Solution:

Q2. The inverse function of , x (-1, 1) , is

7th Jan 2020- Shift II

A

B

D

C


Q3. The range of the function

A

B

D

C


Solution:

Q3. The range of the function

A

B

D

C


A

B

D

Q4. If

C

JEE-Main 2019, 8th April -I


Solution:

A

B

D

Q4. If

C



A B

DC

Q5. If the function f:R-{1,-1}→A defined by , , is surjective, then A is equal to :

R-[-1,0) R-(-1,0)

R-{-1} [0,∞]



#Let’s KILL JEEE

Solution:

Range of y : R-[-1,0) for surjective function, A must be same as above range.

A B

DC

Q5. If the function f:R-{1,-1}→A defined by , , is surjective, then A is equal to :

R-[-1,0) R-(-1,0)

R-{-1} [0,∞]



A

Q6. The domain of the definition of the function is

B

D

C


Solution:

A

Q6. The domain of the definition of the function is

B

D

C


A

Q7. The function

B

D

C

invertible.

injective but not surjective.

surjective but not injective.

neither injective nor surjective.

JEE (Main) 2017


Solution:

A

Q7. The function

B

D

C

invertible.

injective but not surjective.

surjective but not injective.

neither injective nor surjective.

JEE (Main) 2017


A

B

Q8. If 𝛼 is a real number such that 𝛼

C

D

JEE-Main 2019, 10th April -II


Solution:

A B

Q8. If 𝛼 is a real number such that 𝛼

C DJEE-Main 2019, 10th April -II


#Let’s KILL JEEE

Q9. The domain of is:

A

B

(5, ∞)

(4, ∞)

D

C

[4, ∞)

[5, ∞)


Solution:

Q9. The domain of is:

A

B

(5, ∞)

(4, ∞)

D

C

[4, ∞)

[5, ∞)


Q10. Let f : R+ ⟶ R be a function which satisfies then for f (1) =3, f(x) is equal to:

A

B

(2x-1)/x

(x+1)/x

D

C

(2x+1)/x

-(2x+1)/x


Solution:

Q10. Let f : R+ ⟶ R be a function which satisfies then for f(1)=3, f(x) is equal to:

A

B

(2x-1)/x

(x+1)/x

D

C

(2x+1)/x

-(2x+1)/x


Q11. The period of the function

A

B

D

C

2ㄫ

5ㄫ

10ㄫ

8ㄫ


Solution:

Thus LCM of (2𝜋 & 5𝜋) is 10𝜋Hence the Period of f(x) is 10𝜋

Q11. The period of the function

A

B

D

C

2ㄫ

5ㄫ

10ㄫ

8ㄫ


#Let’s KILL JEEE

Homework Questions

A

B

D

C

Q1. Let f : (1,3) ⟶ R be a function defined by , where

[x] denotes the greatest integer ≤ x. Then the range of f is :

Homework Questions

A

Q2. Let f (x) = ax (a > 0) be written as f ( x ) = f1 ( x ) + f2 ( x ) , where f1 ( x ) is an even function and f2 ( x ) is an odd function. Then f1 ( x + y ) + f1( x - y) equals

B

D

C 2f1 ( x + y ) f2( x - y)

2f1 ( x + y ) f1( x - y)

2f1 ( x ) f2 ( y )

2f1 ( x ) f1 ( y )

Homework QuestionsQ3. If g is the inverse of a function f and f’(x) = , then g’(x) is equal to

A

B

D

C

#Let’s KILL JEEE

Thank You

A

B

D

C



Solution of Homework Questions

Solution:

A

B

D

C




A


B

D

C 2f1 ( x + y ) f2( x - y)

2f1 ( x + y ) f1( x - y)

2f1 ( x ) f2 ( y )

2f1 ( x ) f1 ( y )


Solution:

A


B

D

C 2f1 ( x + y ) f2( x - y)

2f1 ( x + y ) f1( x - y)

2f1 ( x ) f2 ( y )

2f1 ( x ) f1 ( y )


Q3. If g is the inverse of a function f and f’(x) = , then g’(x) is equal to

A

B

D

C


Solution:

Q3. If g is the inverse of a function f and f’(x) = , then g’(x) is equal to

A

B

D

C


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JEE Main Functions in One Shot - Amazon Web Services