Relations and Functions

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

Sol:$\begin{array}{l}\begin{array}{l}\begin{array}{l}\begin{array}{l}\hfill \end{array}\end{array}\hfill \end{array}\end{array}$

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

Sol:

$\begin{array}{l}f(x)\frac{1}{2-cosx},\forall x\in R\\ Rangeofcosxis[-1,1]\\ Letf(x)=y=\frac{1}{2-cosx}\\ \Rightarrow 2y-ycosx=1\Rightarrow ycosx=2y-1\\ \Rightarrow cosx=\frac{2y-1}{y}=2-\frac{1}{y}\\ Now-1\le cosx\le 1\\ \Rightarrow -1\le 2-\frac{1}{y}\le 1\Rightarrow -1-2\le -\frac{1}{y}\le 1-2\\ \Rightarrow -3\le -\frac{1}{y}\le -1\Rightarrow 3\ge \frac{1}{y}\ge 1\Rightarrow \frac{1}{3}\le y\le 1\\ \therefore therangeoff=\left[\frac{1}{3},1\right].\end{array}$

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

Sol:

$\begin{array}{}\end{array}Let{x}_1,x_2\in gof$

$gof\{f(x_1)\}=gof\{f(x_2)\}$

$\Rightarrow g(x_1)=g(x_2)\begin{array}{l} \end{array}$

$\therefore$
$\begin{array}{l}{x}_1 =x_2\\ \end{array}$

$\begin{array}{l}Hence,fisone-oneandgisonto.\end{array}$

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

Sol:

$Giventhat,$

$\begin{array}{l}X=\{1,2,3\}andY=\{4,5\}\hfill \end{array}$

$\begin{array}{l}\therefore X *\text{?}Y=\{(1,4),(1,5),\left(2,4\right),\left(2,5\right),(3,4),(3,5)\}\hfill \end{array}$

$\begin{array}{l}\left(i\right)f=\{(1,4),\left(1,5\right),\left(2,4\right),(3,5)\}\hfill \end{array}$

$\begin{array}{l}fisnotafunctionbecausefhasnouniqueimage.\hfill \end{array}$

$\begin{array}{l}\left(ii\right)g=\left\{\left(1,4\right),\left(2,4\right),\left(3,4\right)\right\}\hfill \end{array}$

$\begin{array}{l}Since,gisafunctionaseachelementofthedomainhasauniqueimage.\hfill \end{array}$

$\begin{array}{l}\left(iii\right)h=\{(1,4),(2,5),(3,5)\}\hfill \end{array}$

$\begin{array}{l}Itisafunctionaseachelementofthedomainhasauniqueimage.\hfill \end{array}$

$\begin{array}{l}\left(iv\right)k=\{(1,4),\left(2,5\right)\}\hfill \end{array}$

$\begin{array}{l}kisnotafunctionas3doesnothaveanyimageunderthemapping.\hfill \end{array}$

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

Sol:

$\begin{array}{l}f(z)=\left|z\right|\forall z\in C\\ f(1)=\left|1\right|=1\\ f(-1)=\left|-1\right|=1\\ f(1)=f(-1)\\ But1\ne -1\\ Therefore,itisnotone-ono.\\ Now,letf(z)=y=\left|z\right|.\\ Here,thereisno;pre-imageofnegativenumbers.\\ Hence,itisnotonto.\end{array}$

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

Sol:

$Given that,$
$\begin{array}{l}f = \left\{\left(1,2\right),\left(3,5\right),\left(4,1\right)\right\}\hfill \end{array}$

$and$
$\begin{array}{l}g = \left\{\left(2,3\right),\left(5,1\right),\left(1,3\right)\right\}\hfill \end{array}$

$\begin{array}{l}Now, fog \left(2\right) = f \left\{g\left(2\right)\right\} = f \left(3\right) = 5\hfill \end{array}$

$\begin{array}{l}fog \left(5\right) = f \left\{g\left(5\right)\right\} = f \left(1\right) = 2\hfill \end{array}$

$\begin{array}{l}fog \left(1\right) = f \left\{g\left(1\right)\right\} = f \left(3\right) = 5\hfill \end{array}$

$\begin{array}{l}fog = \left\{\left(2,5\right),\left(5,2\right),\left(1,5\right)\right\}\hfill \end{array}$

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

Sol:

$\begin{array}{l}\left(i\right)Given set of ordered pair is \left\{\left(x,y\right):xisaperson,yisthemotherofx\right\}.\hfill \end{array}$$\begin{array}{l}It represents a function.\hfill \end{array}$

$\begin{array}{l}Here, the image of distinct elements of x under f are not distinct, so it is not an injective but it is a surjective.\hfill \end{array}$

$\begin{array}{l}\left(ii\right)Set of ordered pairs = \left\{\left(a,b\right):aisaperson,bisanancestorofa\right\}\hfill \end{array}$

$\begin{array}{ll}Here, each element of domain does not have a unique image.\hfill & \hfill \end{array}$

$\begin{array}{l}So, it does not represent function.\hfill \end{array}$

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

Sol:

Given that, g = { (1, 1), (2, 3), (3, 5), (4, 7)}.

Here, each element of domain has unique image. So, g is a function.

Now given that, g (x ) = $\alpha$ x + $\beta$

For (1,1) g (1) = $\alpha$ + $\beta$

$\alpha$ + $\beta$ = 1… (i)

For (2,3)   g (2) = 2 $\alpha$ + $\beta$

2 $\alpha$ + $\beta$ = 3 … (ii)

From Equations. (i) and (ii),

2 (1 - $\beta$ ) + $\beta$ = 3

2 - 2 $\beta$ + $\beta$ = 3

2 - $\beta$ = 3

$\beta$ = - 1

If $\beta$ = - 1, then $\alpha$ = 2

$\therefore$ $\alpha$ = 2,  $\beta$ = - 1

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

Sol:

$\begin{array}{l}Given that\hfill \end{array}$$\hfill$$\begin{array}{l}f(x)=x^2- 3x+ 2\hfill \end{array}$ $\begin{array}{ll}f\left\{f(x)\right\} =f\left(x^2-3x+2\right)\hfill & \hfill \end{array}$

$\begin{array}{ll}= {\left(x^2-3x+2\right)}^2- 3\left(x^2-3x+2\right) + 2\hfill & \hfill \end{array}$

$\begin{array}{ll}=x^4+ 9x^2+ 4 - 6x^3- 12x+ 4x^2- 3x^2+ 9x- 6 + 2\hfill & \hfill \end{array}$

$\begin{array}{l}=x^4+ 10x^2- 6x^3- 3x\hfill \end{array}$

$\begin{array}{l}Hence f\left\{f(x)\right\} =x^4- 6x^3+ 10x^2- 3x\hfill \end{array}$

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

Sol:

$\begin{array}{l}Given that, A = \left\{a,b,c,d\right\}\hfill \end{array}$

$\begin{array}{l}and f = \left\{\left(a,b\right),\left(b,d\right),\left(c,a\right),\left(d,c\right)\right\}\hfill \end{array}$

$\begin{array}{l}{f}^{-1}= \left\{\left(b,a\right),\left(d,b\right),\left(a,c\right),\left(c,d\right)\right\}\hfill \end{array}$