#### Provide solution for RD Sharma maths Class 12 Chapter Functions Exercise 2.2 Question 2.

Answer : $f\; o\; g= \left \{ (1,1),(3,1),(4,3),(5,3) \right \}$

$g\; o\; f= \left \{ (3,3),(9,3),(12,9) \right \}$

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 : $f=\left \{ (3,1),(9,3),(12,4) \right \}$

$g=\left \{ (-1,-2),(-2,-4),(-3,-6),(4,8) \right \}$

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

$g\; o\; f=\left \{ (3,3),(9,3),(12,9) \right \}$