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RD Sharma Class 12 Exercise 2.2 Function Solutions Maths - Download PDF Free Online

RD Sharma Class 12 Exercise 2.2 Function Solutions Maths - Download PDF Free Online

Edited By Team Careers360 | Updated on Jan 20, 2022 10:40 AM IST

The RD Sharma Solutions for Class 12 Mathematics help the students to understand and work out the problems efficiently. It is essential to possess the right guide while working on homework or assignments, especially on complex topics like the RD Sharma Class 12th Exercise 2.2. This single compilation is adequate for the CBSE students who are preparing for their board exams.

RD Sharma Class 12 Solutions Chapter 2 Functions - Other Exercise

Functions Exercise 2.2 Question 1 (i) .

Answer :gof(x)=4x2+12x+14andfog=2x2+13
Hint :gof means f(x) is in g(x) function
fog means g(x) function is in f(x) function
Given : f:RRand g:RR
fog:RRand gof:RR
f(x)=2x+3,g(x)=x2+5
Solution :
Now first we find gof
gof(x)=g(2x+3)
=(2x+3)2+5
gof(x)=4x2+12x+9+5 [(a+b)2=a2+b2+2ab]
gof(x)=4x2+12x+14
Now we find fog
fog(x)=f(g(x))
f(x2+5)=2(x2+5)+3
fog(x)=2x2+13
Hence, fog(x)=2x2+13andgof(x)=4x2+12x+14

Then we find fog
fog(x)=f(g(x))
=f(x3)
fog(x)=2x3+x6
Hence, fog=2x3+x6 and gof=(2x+x2)3

Functions Exercise 2.2 Question 1 (ii) .

Answer :fog=2x3+x6andgof=(2x+x2)3
Hint :gof means f(x) function is in g(x) function
fog means g(x) function is in f(x) funcion
Given :f(x)=2x+x2
g(x)=x3
f:RRandg:RR
fog:RRandgof:RR
Solution :
First, we find gof
gof(x)=g(f(x))
=g(2x+x2)
gof(x)=(2x+x2)3
Then we find fog
fog(x)=f(g(x))
=f(x3)
fog(x)=2x3+x6
Hence, fog=2x3+x6 and gof=(2x+x2)3

Functions Exercise 2.2 Question 1 (iii) .

Answer :gof=3(x2+8)3+1andfog=9x6+6x3+9
Hint :gof means f(x) function is in g(x) function
fog means g(x) function is in f(x) function
Given :f(x)=x2+8
g(x)=3x3+1
f:RRandg:RR
fog:RRandgof:RR
Solution :
First, we find gof
Thus, gof(x)=g[f(x)]
gof(x)=g[x2+8]
gof(x)=3[x2+8]3+1
Similarly,
fog(x)=f[g(x)]
fog(x)=f(3x3+1)
fog(x)=[3x3+1]2+8
fog(x)=[9x6+1+6x3]2+8 [(a+b)2=a2+b2+2ab]
fog(x)=9x6+1+6x3+9
Hence, fog=9x6+1+6x3+9andgof=3(x2+8)3+1

Functions Exercise 2.2 Question 1 (iv) .

Answer :fog=|x|andgof=|x|
Hint :gof means f(x) function is in g(x) function
fog means g(x) function is in f(x) function
Given :f(x)=x
g(x)=|x|
Solution :
Since f:RRandg:RR
fog:RRandgof:RR
Now, gof(x)=g(f(x))
=g(x)
gof(x)=|x|
Similarly ,
fof(x)=f(g(x))
=f|x|
fog(x)=|x|
Hence, fog=|x|andgof=|x|

Functions Exercise 2.2 Question 1 (v) .

Answer :fog=9x218x+5fog=9x218x+5andgof=3x2+6x13
Hint :gof means f(x) function is in g(x) function
fog means g(x) function is in f(x) function
Given :f(x)=x2+2x3
g(x)=3x4
Solution :
Since f:RRandg:RR
fog:RRandgof:RR
Now, gof(x)=g(f(x))=g(x)
gof(x)=g(f(x))=g(x2+2x3)
gof(x)=3(x2+2x3)4
gof(x)=3x2+6x94
gof(x)=3x2+6x13
fog(x)=f(g(x))=f(x)
fog(x)=f(g(x))=f(3x4)
fog(x)=(3x4)2+2(3x4)3
=9x2+1624x+6x83 [(ab)2=a2+b22ab]
fog(x)=9x218x+5
Hence, fog=9x218x+5andgof=3x2+6x13
Functions Exercise 2.2 Question 1 (vi).
Hint :gof means f(x) function is in g(x) function
fog means g(x) function is in f(x) function
Given :f(x)=8x3
g(x)=x13
Solution :
Since f:RRandg:RR
fog:RRandgof:RR
Now, gof(x)=g(f(x))=g(8x3)
gof(x)=(8x3)13
=(8)13=(23)13 [as we know,(am)1n=(a)mn]
gof(x)=(8)13(x3)13
gof(x)=2x
Similarly,
fog(x)=f(g(x))=f(x)13
fog(x)=8(x13)3
fog(x)=8x [(am)1n=(a)mn]
Hence, fog=8xandgof=2x

Functions Exercise 2.2 Question 2

Answer :fog={(1,1),(3,1),(4,3),(5,3)}
gof={(3,3),(9,3),(12,9)}
Hint : The set which contains all the elements of all the ordered pairs of relation R is known as the domain of the relation. The set which contains all the second element, is known as the range of the relation.
Given :f={(3,1),(9,3),(12,4)}

g={(1,2),(2,4),(3,6),(4,8)}
Prove :gof and fog are both defined
Solution :f:{3,9,12}{1,3,4}andg:{1,3,4,5}{3,9}
Co-domain of f is a subset of the domain g
So, gof exist and gof:{3,9,12}{3,9}
(gof)(3)=g(f(3))=g(1)=3
(gof)(9)=g(f(9))=g(3)=3
(gof)(12)=g(f(12))=g(4)=9
gof={(3,3),(9,3),(12,9)}
Co-domain of g is subset of the domain of f.
So, fog exist and fog:{1,3,4,5}{3,9,12}
(fog)(1)=f(g(1))=f(3)=1
(fog)(3)=f(g(3))=f(3)=1
(fog)(4)=f(g(4))=f(9)=3
(fog)(5)=f(g(5))=f(9)=3
fog={(1,1),(3,1),(4,3),(5,3)}
Hence proved, gof and fog are both defined.
fog={(1,1),(3,1),(4,3),(5,3)} and
gof={(3,3),(9,3),(12,9)}


Functions Exercise 2.2 Question 3.

Answer :gof={(1,2),(4,4),(9,6),(16,8)}
Hint : The set which contains all the elements of all the ordered pairs of relation R is known as the domain of the relation. The set which contains all the second element is known as the range of the relation.
Given :f={(1,1),(4,2),(9,3),(16,4)}
g={(1,2),(2,4),(3,6),(4,8)}
Prove :gof is defined while fog is not defined.
Solution
Now, Domain of f={1,4,9,16}
Range of f={1,2,3,4}
Domain of g={1,2,3,4}
Range of g={2,4,6,8}
Clearly, Range of f = domain of g
gofis defind.
But Range of g domain of f
fogis not defind.
Now, gof(1)=g(1)=2
gof(4)=g(2)=4
gof(9)=g(3)=6
gof(16)=g(4)=8
gof={(1,2),(4,4),(9,6),(16,8)}
Hence proved, gof is defined but fog is mot defined.
gof={(1,2),(4,4),(9,6),(16,8)}

Functions Exercise 2.2 Question 4.

Answer : gof={(a,a),(b,b),(c,c)}and
fog={(u,u),(v,v),(w,w)}
Hint : For any function to be a bijective, the given function should be one-one and onto.\
Given :A={a,b,c}
B={u,v,w}
Solution :
f and g be two functions from A to B and from B to A ; ABandg:BA
f={(a,v),(b,u),(c,w)}f={(a,v),(b,u),(c,w)}
g={(u,b),(v,a),(w,c)}
For both f and g, Different elements of the domain have different images.
f and f are onto
Again, for each element in co-domain of f and g, there in a preimage in domain
f and f are onto
Thus, f and g are bijectives.
Now,
gof={(a,a),(b,b),(c,c)}and
fog={(u,u),(v,v),(w,w)}
Hence prove, f and g both are bijections
gof={(a,a),(b,b),(c,c)}and fog={(u,u),(v,v),(w,w)}

Functions Exercise 2.2 Question 5.

Answer :fog(2)=633
gof(1)=2188
Hint :gof means f(x) function is in g(x) function
fog means g(x) function is in f(x) function
Given : f:RR;f(x)=x2+8
g:RR;g(x)=3x3+1
Solution :
First, we find fog
fog(x)=f(g(x))=f(3x3+1)
fog(x)=(3x3+1)2+8 [(a+b)2=a2+b2+2ab]
fog(2)=(3×(2)3+1)2+8
=(3×8+1)2+8
=(25)2+8
=625+8
fog(2)=633
Similarly,
gof(x)=g(f(x))=g(x2+8)=3(x2+8)3+1
gof(1)=3(1+8)3+13(9)3+1
=3×729+1
=2188
Hence, fog(2)=633andgof(1)=2188

Function Exercise 2.2 Question 6.

Answer :fog=xandgof=x
Hint :gof means f(x) function is in g(x) function
fog means g(x) function is in f(x) function
Given :f:R+R+ defined by
f(x)=x2
g:R+R+ defined by
g(x)=x
Solution :
fog(x)=f(g(x))=f(x)
fog(x)=(x)2=x
Similarly,
gof(x)=g(f(x))=g(x2)
gof(x)=(x)2=x
Thus
fog(x)=gof(x)
x=x
Hence proved functions are equal.
fog=xandgof=x

Function Exercise 2.2 Question 7.

Answer : foggof
Hint :gof means f(x) function is in g(x) function
fog means g(x) function is in f(x) function
Given :f:RR defined by
f(x)=x2
g:RR defined by
g(x)=x+1
Solution :
fog(x)=f(g(x))=f(x+1)
fog(x)=(x+1)2 [(a+b)2=a2+b2+2ab]
fog(x)=x2+1+2x ......(i)
Similarly,
gof(x)=g(f(x))=g(x2)
gof(x)=x2+1 .....(ii)
Now we have to make a comparison between eq. (i) and (ii)
fog(x)gof(x)
x2+1+2xx2+1
Hence proved foggof

Function Exercise 2.2 Question 8.

Answer :fog=gof=IR
Hint :gof means f(x) function is in g(x) function
fog means g(x) function is in f(x) function
Given :f:RR and g:RR are defined by
f(x)=x+1g(x)=x1
Solution :
fg(x)=f(g(x))=f(x1)fg(x)=x1+1fg(x)=x=IR .....(i)
Similarly,
gf(x)=g(f(x))=g(x+1)gf(x)=x+11gf(x)=x=IR ..... (ii)
From equation (i) and (ii)
fog=gof=IR
Hence proved : fog=gof=IR

Function Exercise 2.2 Question 9.

Answer :ho(gof)=(hog)of
Hint :gof means f(x) function is in g(x) function
fog means g(x) function is in f(x) function
For associative property ho(gof)=(hog)of
Given :
f:NZ0,:Z0Q,R:QRf(x)=2xg(x)=1xh(x)=ex
Solution :
gf:NQ and hg:Z0Rh(gf):NR and (hg)f:NR
So, both have same domains [sincef(x)=2x]
(gf)(x)=g[f(x)]=g(2x)=12x(hg)(x)=h[g(x)]=h(1x)e1x .....(ii)
Now,
[h(gf)](x)=h[(gf)(x)]=h(12x) ....[From (i)]
=e12x ....[From (ii)]
[(hg)f(x)]=(hg)[f(x)]
=(hg)(2x) ...[since f(x)=2x]
=e12x ....[From (ii)]
[h(gf)](x)=[(hg)f(x)],N
So, h(gf)=(hg)f
Hence, the associative property has benn verified.

Function Exercise 2.2 Question 10.

Answer : ho(gof)=(hog)of
Hint :gof means f(x) function is in g(x) function
fog means g(x) function is in f(x) function
Given : f:NN,g:NN,h:NR defined as
f(x)=2xg(y)=3y+4h(z)=sinz All x,y,zN
Solution :
h(gf)(x)=h(gf(x))=h(g(f(x)))=h(g(2x))=h(3(2x)+4)=h(6x+4)=sin(6x+4)xN
Similarly,
((hg)f)(x)=(hg)(f(x))=(hg)(2x)=h(g(2x))=h(3(2x)+4)
=h(6x+4)=sin(6x+4)xϵN
This shows, ho(gof)=(hog)of

Functions Exercise 2.2 Question 11.

Answer :
f:NN by f(x)=x+1g:NN by g(x)={x1 if x>1,1 if x=1
Hint : Onto functions means every range has some preimage in the domain of function.
Given :f:NNandg:NN
Solution :
Let f:NNbef(x)=x+1
and g:NNbeg(x)={x1,x>11,x=1
We will first show that f is not onto
Checking f is not onto
f:NN be f(x)=x+1y=f(x), where yNy=x+1;x=y1
for y=1;x=11=0
But 0 is not a natural number
f is not onto.
Finding gof:
f(x)=x+1g(x)={x1x>11x=1f(x)=x+1;g(x)=1
Since g(x)=1,g(f(x))=1
So, gof=1
For x>1
Since,
g(x)=x1,g(f(x))=f(x)1gf=(x+1)1gf=x
So, gf={x,x>11,x=1}
Let gf=y, where yN
So, y={x,x>11,x=1}
Here, y is a natural number. As y=x
So, x is also a natural number.
Hence, gof is onto.

Functions Exercise 2.2 Question 12.

Answer :f(x)=xandg(x)=|x|
where f:NZandg:ZZ
Hint : An injective function is a function that maps distinct elements of its domain to the distinct elements of its co-domain.
Given : f:NZandg:ZZ
Solution :
Let f(x)=xandg(x)=|x|
g(x)=|x|={xx0xx<0
Checking g(x) injective (one-one)
g(1)=|1|=1
g(1)=|1|=1
Since, different elements 1, -1 have the same image 1
g is not injective (one -one).
Checking for injective (one-one)
f:NZ&g:ZZf(x)=x and g(x)=|x|gf(x)=g(f(x))=|f(x)|=|x|={x,x0x,x<0} Here gf(x):NZ
So, x is always a natural number.
Here |x| will always be a natural number.
So, gof(x) has a unique image
gof(x) is injective.
Hence, gof is injective but g is not injective

Functions Exercise 2.2 Question 13.

Answer :gof is a one -one function.
Hint : one-one functions means every domain has a distinct image. If one-one is given for any function
f(x) as if F(x1)=F(x2) then x1=x2 where x1,x2 domains of f(x)
Given : f:ABandg:BC one-one function
Solution :
 Let, x,yA such that gf(x)=gf(y)g(f(x))=g(f(y))
f(x)=f(y) [ g is one - one ]
x=y [ f is one - one]
gof is a one - one function

Functions Exercise 2.2 Question 14.

Answer :gof is an onto function
Hint : Onto function means every range has some pre-image in the domain of function.
Given :f:ABandg:BC are onto function
Prove :gof:AC is an onto function
Solution :
Let yϵC then
gf(x)=yg(f(x))=y ....(i)
Since, g is onto, for each element in C, then exists a pre-image in B.
g(x)=y .....(ii)
From (i) and (ii)
f(x)=x
Since f is onto, for each element in B, there exist a pre image in A
f(x)=x ....(iii)
From (ii) and (iii) we can conclude that for each yϵC, there exist a preimage in A
Such that gof(x)=y
gof is onto.


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The 2nd chapter in RD Sharma Class 12th Exercise 2.2 is regarding Functions in mathematics. This chapter focuses on the various kinds of functions, compositions of functions, and inverse of functions. Again, this is a specific topic to be grasped by students compared to the other chapters in the book. The first exercise, 2.1, is a bit essential in the mathematical functions. The second exercise, 2.2, dives deeply into the concepts of fog and gof, injective, surjective, and bijective.

The Class 12 RD Sharma Chapter 2 Exercise 2.2 Solution will lend the students a helping hand. Making the students understand the concept of functions is essential as the upcoming chapters are based on it. For the convenience of the students, the solutions for RD Sharma Class 12th Exercise 2.2 are given in the same order as present in the textbooks.

RD Sharma Class 12th Exercise 2.2 is divided into two levels, Level 1 and Level 2, according to the questions' complexity and difficulty. Level 1 has 21 questions, and level 2 has a couple of questions. Even though numerous examples are provided in the book, solutions elaborately make the students understand the concept more easily and effectively. Therefore, every answer provided in the RD Sharma Class 12 Solutions Chapter 2 Ex 2.2 is given in a more manageable format for the welfare of the students.

With the help of this best solutions book, the students would find the Function chapter easier than before. The RD Sharma Class 12 Solutions Function Ex 2.2 makes the students connect more effortlessly with the concepts than expected. Moreover, once they start using it to prepare their homework and assignments, facing tests and exams in this chapter would become easier based on the NCERT syllabus. Hence, if you are a CBSE board class 12 student, it is better to possess the RD Sharma Class 12th Exercise 2.2 Solutions as there are many chances of questions being asked from this book on your board exam.

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