Relations and Functions

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l}\begin{array}{l}f\left(x\right) = 2x - 3\hfill \end{array}\end{array}$

$\begin{array}{l}\begin{array}{l}Let f\left(x\right)= y = 2x – 3\hfill \end{array}\\ \Rightarrow y+3=2x\Rightarrow x=\frac{y\text{?}+3}{2}\\ \therefore f^{-1}(y)=\frac{y+3}{2}0r{f}^{-1}(x)=\frac{x+3}{2}\end{array}$

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l}f(x)=2x+1andg(x)=x^2-2\\ \therefore gof=g\left[f(x)\right]\\ ={\left[2x+1\right]}^2-2\\ =4x^2+4x+1-2\\ =4x^2+4x-1\\ Hence,gof=4x^2+4x-1.\end{array}$

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol:

This is a Short Answer Type Question as classified in NCERT Exemplar

Sol: R = { (a, a), (b, c), (a, b)}.

To make R as reflexive we must add (b, b) and (c, c) to R. Also, to make R as transitive we must add

(a, c) to R.

So, minimum number of ordered pair is to be added are (b, b), (c, c), (a, c).

This is a Long Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l}(i):Giventhat\\ a\ast b=1+ab\forall a,b\in R\\ andb\ast a=1+ba\forall a,b\in R\\ a\ast b=b\ast a=1+ab\\ So,\ast iscommutative.\\ Nowa\ast (b\ast c)=(a\ast b)\ast c\forall a,b,c\in R\\ L.H.S.a\ast (b\ast c)=a\ast (1+bc)=1+a(1+bc)=1+a+abc\\ R.H.S.(a\ast b)\ast c=(1+ab)\ast c=1+(1+ab).c=1+c+abc\\ So,\ast isnotassociative.\\ Hence,\ast iscommutativebutnotassociative.\end{array}$

This is a Long Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l}(i)a\ast b=a-b\in Qa,b\in Q,\\ So,\ast isbinaryoperation.\\ a\ast b=a-bandb\ast a=b-a\forall a,b\in Q\\ a-b\ne b-a\\ So,\ast isnotcommutative.\\ (ii)a\ast b={{\displaystyle a}}^2+{{\displaystyle b}}^2\in Q,So,\ast isbinaryoperation.\\ a\ast b=b\ast a\\ \Rightarrow a^2+b^2=b^2+a^2\forall a,b\in Q\\ Whichistrue.So,\ast iscommutative.\\ (iii)a\ast b=a+ab\in Q,So,\ast isbinaryoperation.\\ a\ast b=a+abandb\ast a=b+ba\\ a+ab\ne b+ba\Rightarrow a\ast b\ne b\ast a\forall a,b\in Q.\\ So,\ast isnotcommutative.\\ (iv)a\ast b={{\displaystyle (a-b)}}^2\in Q,So,\ast isbinaryoperation.\\ a\ast b={{\displaystyle (a-b)}}^{2}andb\ast a={{\displaystyle (b-a)}}^2\\ a\ast b=b\ast a\Rightarrow {{\displaystyle (a-b)}}^2={{\displaystyle (b-a)}}^2\forall a,b\in Q.\\ So,\ast iscommutative.\end{array}$

This is a Long Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l}(i)fog\Rightarrow f\left[\left(g(x\right)\right]=[g]{(x)}^2+3\left[\left(g(x\right)\right]+1\\ ={(2x-3)}^2+3(2x-3)+1\\ =4x^2+9-12x+6x-9+1=4x^2-6x+1\\ (ii)gof\Rightarrow g\left[f(x)\right]=2\left[{{\displaystyle x}}^2+3x+1\right]-3\\ =2{{\displaystyle x}}^2+6x+2-3=2{{\displaystyle x}}^2+6x+1\\ (iii)fof\Rightarrow f\left[\left(f(x\right)\right]={{\displaystyle \left[f(x)\right]}}^2+3\left[f(x)\right]+1\\ =({{\displaystyle {{\displaystyle x}}^2+3x+1)}}^2+3({{\displaystyle x}}^2+3x+1)+1\\ ={{\displaystyle x}}^4+9{{\displaystyle x}}^2+1+6{{\displaystyle x}}^3+6x+2{{\displaystyle x}}^2+3{{\displaystyle x}}^2+9x+3+1\\ ={{\displaystyle x}}^4+6{{\displaystyle x}}^3+14{{\displaystyle x}}^2+15x+5\\ (iv)gog\Rightarrow g\left[g(x)\right]=2\left[g(x)\right]-3=2(2x-3)-3=4x-6-3=4x-9\end{array}$

This is a Long Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Afunctionf:X\to Yissaidtobeinvertibleifthereexistsa\\ functiong:Y\to Xsuchthatgof={{\displaystyle I}}_{X}andfog={{\displaystyle I}}_{Y}andthenthe\\ inverseoffisdenotedby{{\displaystyle f}}^{-1}.\\ Afunctionf:X\to Yissaidtobeinvertibleifffisabijectivefunction.\end{array}$

This is a Long Answer Type Question as classified in NCERT Exemplar

Sol:

$\begin{array}{l}Here,A=\left\{1,2,3,\dots ,9\right\}\\ andR\to A\times Adefinedby(a,b)R(c,d)\Rightarrow a+d=b+c\\ \forall (a,b),(c,d)\in A\times A\\ Forreflexive:(a,b)R(a,b)=a+b=b+a\forall a,b\in Awhichistrue.So,Risreflexive.\\ Forsymmetric:(a,b)R(c,d)=(c,d)R(a,b)\\ L.H.S.a+d=b+c\\ R.H.S.c+b=d+a\\ L.H.S.=R.H.S.So,Rissymmetric.\\ Fortransitive:(a,b)R(c,d)=(c,d)R(e,f)\iff (a,b)R(e,f)\\ \Rightarrow a+d=b+candc+f=d+e\\ \Rightarrow a+d=b+candd+e=c+f\\ \Rightarrow (a+d)-(d+e)=(b+c)-(c+f)\\ \Rightarrow a-e=b-f\\ \Rightarrow a+f=b+e\\ \Rightarrow (a,b)R(e,f)\\ So,Ristransitive.\\ Hence,Risanequivalencerelation.\\ Equivalentclassof\left\{\left(2,5\right)\right\}is\left\{\left(1,4\right),\left(2,5\right),\left(3,6\right),\left(4,7\right),\left(5,8\right),\left(6,9\right)\right\}\end{array}$

This is a Long Answer Type Question as classified in NCERT Exemplar

Sol: