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.
Functions Exercise 2.2 Question 1 (i) .
Answer :$g\; o\; f \left ( x \right )=4x^{2}+12x+14 \; \text {and}\; f\; o\; g\; =2x^{2}+13$Functions Exercise 2.2 Question 1 (ii) .
Answer :$f\; o\; g=2x^{3}+x^{6}\; \text {and}\; g\; o\; f=\left ( 2x+x^{2} \right )^{3}$Functions Exercise 2.2 Question 1 (iii) .
Answer :$g\; o\; f = 3\left ( x^{2}+8 \right )^{3}+1\; \text {and}\; f\; o\; g=9x^{6}+6x^{3}+9$Functions Exercise 2.2 Question 1 (iv) .
Answer :$f\; o\; g=\left | x \right | \; \text {and} \; g\; o\; f=\left | x \right |$Functions Exercise 2.2 Question 1 (v) .
Answer :$f\; o\; g = 9x^{2}-18x+5$$f\; o\; g = 9x^{2}-18x+5\; \text {and}\; g\; o\; f=3x^{2}+6x-13$Functions Exercise 2.2 Question 2
Answer :$f\; o\; g= \left \{ (1,1),(3,1),(4,3),(5,3) \right \}$$g=\left \{ (-1,-2),(-2,-4),(-3,-6),(4,8) \right \}$
Prove :$g\; o\; f$ and $f\; o\; g$ are both defined
Solution :$f:\left \{ 3,9,12 \right \}\rightarrow \left \{ 1,3,4 \right \}\; \text {and}\; g:\left \{ 1,3,4,5 \right \}\rightarrow \left \{ 3,9 \right \}$
Co-domain of f is a subset of the domain g
So, $g\; o\; f$ exist and $g\; o\; f:\left \{ 3,9,12 \right \}\rightarrow \left \{ 3,9 \right \}$
$(g\; o\; f)(3)=g(f(3))=g(1)=3$
$(g\; o\; f)(9)=g(f(9))=g(3)=3$
$(g\; o\; f)(12)=g(f(12))=g(4)=9$
$g\; o\; f =\left \{ \left ( 3,3 \right ),\left ( 9,3 \right ),\left ( 12,9 \right ) \right \}$
Co-domain of g is subset of the domain of f.
So, $f\; o\; g$ exist and $f\; o\; g:\left \{ 1,3,4,5 \right \}\rightarrow \left \{ 3,9,12 \right \}$
$(f\; o\; g)(1)=f(g(1))=f(3)=1$
$(f\; o\; g)(3)=f(g(3))=f(3)=1$
$(f\; o\; g)(4)=f(g(4))=f(9)=3$
$(f\; o\; g)(5)=f(g(5))=f(9)=3$
$f\; o\; g=\left \{ (1,1),(3,1),(4,3),(5,3) \right \}$
Hence proved, $g\; o\; f$ and $f\; o\; g$ are both defined.
$f\; o\; g=\left \{ (1,1),(3,1),(4,3),(5,3) \right \}$ and
$g\; o\; f=\left \{ (3,3),(9,3),(12,9) \right \}$
Functions Exercise 2.2 Question 3.
Answer :$g\; o\; f = \left \{ \left ( 1,2 \right ),\left ( 4,-4 \right ),\left ( 9,-6 \right ),\left ( 16,8 \right ) \right \}$Functions Exercise 2.2 Question 4.
Answer : $g\; o\; f=\left \{ (a,a),(b,b),(c,c) \right \}\; \text {and}$Functions Exercise 2.2 Question 5.
Answer :$f\; o\; g (2)=633$Function Exercise 2.2 Question 6.
Answer :$f\; o\; g=x \; \text {and}\; g\; o\; f=x$Function Exercise 2.2 Question 7.
Answer : $f\; o\; g \neq g\; o\; f$Function Exercise 2.2 Question 8.
Answer :$f\; o\; g=g\; o\; f=I_{R}$Function Exercise 2.2 Question 9.
Answer :$h\; o(g\; o\; f)=(h\; o\; g)o\; f$Function Exercise 2.2 Question 10.
Answer : $h\; o(g\; o\; f)=(h\; o\; g)o\; f$Functions Exercise 2.2 Question 11.
Answer :Functions Exercise 2.2 Question 12.
Answer :$f(x)=x \; \text {and}\; g(x)=\left | x \right |$Functions Exercise 2.2 Question 13.
Answer :$g\; o\; f$ is a one -one function.Functions Exercise 2.2 Question 14.
Answer :$g\; o\; f$ is an onto functionMathematics may seem complicated for some students, but with the help of RD Sharma Solutions for Class 12 Mathematics, they can figure out an easier way to solve the problems.
Not every child could understand a standard way of finding the answer; hence, this book gives multiple formats of arriving at mathematics solutions. And the most significant advantage is that you can find this book free of cost, without wanting you to pay even a penny.
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.
Chapter-wise RD Sharma Class 12 Solutions
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