Relations and Functions Class 11 Notes: CBSE Maths Chapter 2

1. Introduction:

• In this chapter, we'll learn how to link pairs of objects from two sets to form a relation between them. 

• We'll see how a relation can be classified as a function. 

• Finally, we'll look at several types of functions, as well as some standard functions.

2. Relations:

2.1 Cartesian Product of Sets

Definition:

• Given two non-empty sets  $ P $  and  $ Q $  . 

• The Cartesian product  $ P\times Q $  is the set of all ordered pairs of elements from  $ P $  and  $ Q $  that is

• \[\text{P }\!\!\times\!\!\text{ Q =  }\!\!\{\!\!\text{ }\left( \text{p, q} \right)\text{ ; p}\in \text{P ; q}\in \text{Q }\!\!\}\!\!\text{ }\]

2.2 Relation:

2.2.1 Definition: 

• Let  $ A $  and  $ B $  be two non-empty sets. 

• Then any subset ‘\[\text{R}\]’ of \[\text{A }\!\!\times\!\!\text{ B}\] is a relation from  $ A $ and  $ B $ . 

• If\[\left( \text{a, b} \right)\in ~\text{R}\] , then we can write it as a  $ \text{R b} $  which is read as  $ \text{a} $  is related to \[\text{b}\]  ‘by the relation \[\text{R}\]’, ‘ $ \text{b} $ ’ is also called image of ‘ $ \text{a} $ ’ under $ \text{R} $ .

2.2.2 Domain and Range of a Relation: 

• If  $ \text{R} $  is a relation from  $ A $  to  $ B $ , then the set of first elements in  $ \text{R} $  is known as domain & the set of second elements in  $ \text{R} $  is called range of  $ \text{R} $  symbolically.

• Domain of \[\text{R =  }\!\!\{\!\!\text{  x:}\left( \text{x, y} \right)\in \text{R }\!\!\}\!\!\text{ }\] 

• Range of \[\text{R =  }\!\!\{\!\!\text{  y:}\left( \text{x, y} \right)\in \text{R }\!\!\}\!\!\text{ }\] 

• The set  $ B $  is considered as co-domain of relation  $ \text{R} $ . 

• Note that,

$ \text{range }\!\!~\!\!\text{ }\subset \text{co-domain} $ 

Note : 

Total number of relations that can be defined from a set  $ A $  to a set  $ B $   is the number of possible subsets of \[\text{A }\!\!\times\!\!\text{ B}\]. 

If  \[\text{n}\left( \text{A} \right)\text{ = p}\] and \[\text{n}\left( \text{B} \right)\text{ = q}\], then 

$\text{n}\left( \text{A }\times \text{ B} \right)\text{ = pq }~\text{and}$ total  number of relations is  $ {{\text{2}}^{\text{pq}}} $  .

2.2.3 Inverse of a Relation: 

• Let  $ \text{A, B} $   be two sets and let  $ \text{R} $  be a relation from a set  $ A $  to set  $ B $ . Then the inverse of  $ \text{R} $ denoted as  $ {{\text{R}}^{\text{-1}}} $ , is a relation from  $ B $  to  $ A $   and is defined by

\[{{\text{R}}^{\text{--1}}}\text{ =  }\!\!\{\!\!\text{ }\left( \text{b, a} \right)\text{:}\left( \text{a, b} \right)\in \text{R }\!\!\}\!\!\text{ }\] 

• Clearly 

\[\left( \text{a, b} \right)\in \text{R}\Leftrightarrow \left( \text{b, a} \right)\in {{\text{R}}^{\text{--1}}}\] 

• Also, 

$ \text{Dom}\left( \text{R} \right)\text{ = Range(}{{\text{R}}^{\text{--1}}}\text{)  }\!\!~\!\!\text{ and}$

$\text{Range}\left( \text{R} \right)\text{ = Dom(}{{\text{R}}^{\text{--1}}}\text{)}$

3. Functions:

3.1 Definition: 

A relation ‘ $ \text{f} $ ’ from a set  $ A $  to set  $ B $  is said to be a function if every element of set  $ A $  has one and only one image in set  $ B $ .

Notations:

Notations

Notations 2

Notations 3

3.2 Domain, Co-domain and Range of a function:

Domain:

The domain is believed to be the biggest set of  $ \text{x -} $  values for which the formula provides real  $ \text{y -} $  values when \[\text{y = f}\left( \text{x} \right)\]  is defined using a formula and the domain is not indicated explicitly.

The domain of   $ y=f(x) $  is the set of all real  $ x $  for which  $ f(x) $   is defined (real).

Rules for finding Domain:

1. Even roots (square root, fourth root, etc.) should have non–negative expressions.

2. Denominator  $ \ne 0 $  

3. \[\text{lo}{{\text{g}}_{\text{a}}}\text{x}\] is defined when \[\text{x  0, a  0}\] and \[\text{a}\ne 1\] 

4. If domain \[\text{y = f(x)}\] and \[\text{y = g(x)}\] are  $ {{D}_{1}} $  and  $ {{D}_{2}} $  respectively then the domain of  $ f(x)\pm g(x) $  or  $ f(x).g(x) $  is  $ {{D}_{1}}\cap {{D}_{2}} $ . 

While domain of   $ \dfrac{f(x)}{g(x)} $  is  $ {{D}_{1}}\cap {{D}_{2}}-\left\{ x:g(x)=0 \right\} $  

Range: 

The set of all  $ \text{f -} $  images of elements of  $ A $  is known as the range of   $ f $  and can be denoted as  $ f(A) $ .

$ \text{Range}=f(A)=\left\{ f(x):x\in A \right\} $  

$ \text{f(A)}\subseteq \text{B} $  {Range \[\subseteq \]Co-domain}

Rule for Finding Range: 

First of all find the domain of  $ y=f(x) $ 

i. If domain  $ \in  $  finite number of points  $ \Rightarrow  $  range  $ \in  $  set of corresponding  $ f(x) $  values.

ii. If domain  $ \in R $  or  $ R- $  {Some finite points}

Put  $ y=f(x) $ 

Then express  $ x $   in terms of  $ y $ .From this find  $ y $  for  $ x $  to be defined. (i.e., find the values of  $ y $  for which  $ x $  exists).

iii. If domain  $ \in  $  a finite interval, find the least and greater value for range using monotonicity.

Note:

1. Question of format:

$ \left( y=\dfrac{Q}{Q};y=\dfrac{L}{Q};y=\dfrac{Q}{L} \right)\text{ } $  

$ Q\to  $  Quadratic

$ L\to  $  Linear

Range is found out by cross-multiplying & creating a quadratic in  $ 'x' $   & making \[D\ge 0\text{ }(\text{as }x\in R)\]

2. Questions to determine the range of values in which the given expression  $ y=f(x) $  can be converted into  $ x $   (or some function of   $ x= $   expression in ‘ $ y $ ’ . 

Do this & apply method (ii) .

3. Two functions  $ f $  &  $ g $  are said to be equal if

1. Domain of  $ f= $   Domain of  $ g $  

2. Co-domain of  $ f= $ Co-domain of  $ g $  

3.  $ f(x)=g(x) $   $ \forall x\in  $  Domain

3.3 Kinds of Functions:

Kinds of functions

Note:

• Injective functions are called as one-to-one functions.

• Surjective functions are also known as onto functions.

• Bijective functions are also known as (one-to-one) and (onto) functions.

Relations Which Cannot be Categorized as a Function:

Relations which cannot be Categorized as a Function (a)

As not all elements of set $ A $  are associated with some elements of set  $ B $  . (Violation of– point  $ (i) $  – definition  $ 2.1 $ )

Relations which cannot be Categorized as a Function (b)

An element of set  $ A $  is not associated with a unique element of set  $ B $ , (violation of point  $ (ii) $   definition  $ 2.1 $ )

Methods to check one-one mapping:

1. Theoretically: 

$ f({{x}_{1}})=f({{x}_{2}}) $  

$ \Rightarrow {{x}_{1}}={{x}_{2}} $  

then  $ f(x) $  is one-one.

2. Graphically: 

A function is one-one, if no line parallel to  $ x-axis $  meets the graph of function at more than one point.

3. By Calculus: 

For checking whether  $ f(x) $  is One-One, find whether function is only increasing or only decreasing in their domain. If yes, then function is one-one, that is if  $ f'(x)\ge 0,\forall x\in  $  domain or, if  $ f'(x)\ge 0,\forall x\in  $  domain, then function is one-one.

3.4 Some Standard Real Functions & their Graphs:

3.4.1 Identity Function: 

The function  $ f:R\to R $   defined by 

$y=f\left( x \right)=x\forall x\in R$  is called identity function.

Identity Function

3.4.2 Constant Function: 

The function  $ f:R\to R $  defined by 

\[y=f\left( x \right)=c,\forall x\in R\] 

Constant Function

3.4.3 Modulus Function: 

The function  $ f:R\to R $  defined by 

$f(x)=\left\{\begin{array}{cc}x ; & x \geq 0 \\ -x ; & x<0\end{array}\right.$ is called modulus function. It is denoted by 

$ y=f(x)=\left| x \right|$ 

Absolute value function

It is also known as “Absolute value function”.

Properties of Modulus Function: 

The modulus function has the following properties:

1. For any real number  $ x $ , we have  $ \sqrt{{{x}^{2}}}=\left| x \right| $ 

2. $ \left| xy \right|=\left| x \right|\left| y \right| $  

3. $ \left| x+y \right|\le \left| x \right|+\left| y \right| $    Triangle inequality

4. $ \left| x-y \right|\ge \left| \left| x \right|-\left| y \right| \right| $    Triangle inequality

 3.4.4 Signum Function:

The function  $ f:R\to R $  define by 

$f(x)=\left\{\begin{array}{l}1: x>0 \\ 0: x=0 \\ -1: x<0\end{array}\right.$ is called signum function.

It is usually denoted as  $y=f(x)=\operatorname{sgn}(x) \mid$ 

Signum Function

Note:

$\operatorname{sgn}(x)= \begin{cases}\frac{|x|}{x} ; & x \neq 0 \\ 0 ; & x=0\end{cases}$

3.4.5 Greatest Integer Function: 

The function  $ f:R\to R $  defined as the greatest integer less than or equal to  $ x $  . 

It is usually denoted as \[y=f\left( x \right)=\left[ x \right]\].

Greatest Integer Function

Properties of Greatest Integer Function: 

If  $ n $  is an integer and  $ x $  is any real number between  $ n $  and   $ n+1 $  , then the greatest integer function has the following properties:

1.  $ \left[ -n \right]=-\left[ n \right] $  

2.  $ \left[ x+n \right]=\left[ x \right]+n $  

3.  $ \left[ -x \right]=\left[ x \right]-1 $  

4. $[x]+[-x]=\left\{\begin{array}{l}-1, \text { if } x \notin I \\ 0, \text { if } x \in I\end{array}\right.$

Note:

Fractional part of  $ x $ , denoted by  $ \left\{ x \right\} $  is given by \[x\left[ x \right]\] , Hence

$\{x\}=x-[x]=\left\{\begin{array}{cc}x-1 ; & 1 \leq x<2 \\ x & 0 \leq x \leq 1 \\ x+1 & -1 \leq x<0\end{array}\right.$

3.4.6  Exponential Function:

\[f(x)={{a}^{x}}\], 

\[a > 0,a\ne 1\] 

Domain:  $ x\in R $  

Range:  $ f(x)\in \left( 0,\infty  \right) $  

Exponential Function a

Exponential Function b

3.4.7  Logarithm Function:

$ f(x)={{\log }_{a}}x $  ,

$ a > 0,a\ne 1 $  

Domain: $ x\in \left( 0,\infty  \right) $  

Range: $ y\in R $  

Logarithm Function (a)

Logarithm Function (b)

a) The Principal Properties of Logarithms:

Let  $ M $  and  $ N $  be the arbitrary positive numbers,\[\text{a  0, a}\ne 1\text{, b  0, b}\ne 1\] 

1. $ {{\log }_{b}}a=a\text{      }\Rightarrow a={{b}^{c}} $  

2. $ {{\log }_{a}}\left( M.N \right)={{\log }_{a}}M+{{\log }_{a}}N $  

3. $ {{\log }_{a}}\left( \dfrac{M}{N} \right)={{\log }_{a}}M-{{\log }_{a}}N $  

4. $ {{\log }_{a}}{{M}^{N}}=N{{\log }_{a}}M $  

5. \[{{\log }_{b}}a=\dfrac{{{\log }_{c}}a}{{{\log }_{c}}b},c > 0,c\ne 1\] 

6. $\begin{equation} a^{\log _{c} b}=b^{\log _{c} a}, a, b, c>0, c \neq 1\end{equation}$

Note:

• $ \text{lo}{{\text{g}}_{\text{a}}}\text{a = 1} $  

• $ \text{lo}{{\text{g}}_{\text{b}}}\text{a }\text{. lo}{{\text{g}}_{\text{c}}}\text{b }\text{. lo}{{\text{g}}_{\text{a}}}\text{c = 1} $  

• $ \text{lo}{{\text{g}}_{\text{a}}}\text{1 = 0} $  

• $ {{\text{e}}^{\text{xlna}}}\text{ = }{{\text{e}}^{\text{xln}{{\text{a}}^{\text{x}}}}}\text{ = }{{\text{a}}^{\text{x}}} $  

b) Properties of Monotonicity of Logarithm:

1. If \[\text{a   1, lo}{{\text{g}}_{\text{a}}}\text{x  lo}{{\text{g}}_{\text{a}}}\text{y       }\Rightarrow 0 < x < \text{y}\] 

2. If  $ \text{0  a  1, lo}{{\text{g}}_{\text{a}}}\text{x  lo}{{\text{g}}_{\text{a}}}\text{y   }\Rightarrow \text{x  y  0} $  

3. If  $ \text{a  1} $  then  $ \text{lo}{{\text{g}}_{\text{a}}}\text{x  p     }\Rightarrow \text{0  x  }{{\text{a}}^{\text{p}}} $  

4. If  $ \text{a  1} $  then  $ \text{lo}{{\text{g}}_{\text{a}}}\text{x  p     }\Rightarrow \text{x  }{{\text{a}}^{\text{p}}} $  

5. If  $ \text{0  a  1} $  then  $ \text{lo}{{\text{g}}_{\text{a}}}\text{x  p      }\Rightarrow \text{x  }{{\text{a}}^{\text{p}}} $  

6. If  $ \text{0  a  1} $  then  $ \text{lo}{{\text{g}}_{\text{a}}}\text{x  p     }\Rightarrow \text{0  x  }{{\text{a}}^{\text{p}}} $  

Note:

• The logarithm is positive if the exponent and base are on the same side of unity.

• The logarithm is negative if the exponent and base are on opposite sides of unity.

4. Algebra of Real Function:

We'll learn how to add two real functions, remove one from another, multiply a real function by a scalar (a scalar is a real integer), multiply two real functions, and divide one real function by another in this part.

 4.1 Addition of Two Real Functions:

Let  $ f:X\to R $  and  $ g:X\to R $  by any two real functions, where  $ x\subset R $ . Then, we define  $ \left( f+g \right):X\to R $  by 

$ \left( f+g \right)\left( x \right)=f\left( x \right)+g\left( x \right) $  for all  $ x\in X $ .

4.2 Subtraction of a Real Function from another:

Let  $ f:X\to R $ be any two any two real functions, where  $ x\subset R $ .

Then, we define  $ \left( f-g \right):X\to R $  by 

$ \left( f-g \right)\left( x \right)=f\left( x \right)-g\left( x \right) $  for all  $ x\in X $ .

4.3 Multiplication by a Scalar:

Let  $ f:X\to R $  be a real valued function and  $ \alpha  $  be a scalar. 

Here by scalar, we mean a real number. 

Then the product  $ \alpha f $  is a function from  $ X $  to  $ R $  defined as  $ \left( \alpha f \right)\left( x \right)=\alpha f\left( x \right),x\in X $ .

4.4 Multiplication of Two Real Functions:

The product (or multiplication) of two real functions  $ f:X\to R $ and  $ g:X\to R $ is a function  $ fg:X\to R $  defined as

$ \left( fg \right)\left( x \right)=f\left( x \right)g\left( x \right) $  for all  $ x\in X $ .

This is also known as pointwise multiplication.

4.5 Quotient of Two Real Functions:

Let  $ f $  and  $ g $  be two real functions defined from  $ X\to R $  where  $ X\subset R $ . 

The quotient of  $ f $  by  $ g $  denoted by  $ \dfrac{f}{g} $  a is a function defined as  $ \left( \dfrac{f}{g} \right)\left( x \right)=\dfrac{f\left( x \right)}{g\left( x \right)} $  , 

Provided   $ g\left( x \right)\ne 0,x\in X $ .

 4.6 Even and Odd Functions

• Even Function: 

i. \[f\left( -x \right)=f\left( x \right),\forall x\in \] Domain

ii. The graph of an even function  $ y=f\left( x \right) $  is symmetric about the  $ y- $  axis, that is  $ \left( x,y \right) $  lies on the graph  $ \Leftrightarrow \left( -x,y \right) $  lies on the graph.

Even Function

• Odd Function: 

i. $ f\left( x \right)=-f\left( x \right),\forall x\in  $  Domain

ii. The graph of an odd function  $ y=f\left( x \right) $ is symmetric about origin that is if point  $ \left( x,y \right) $  is on the graph of an odd function, then  $ \left( -x,-y \right) $  will also lie on the graph.

Odd Function

5. Periodic Function

• Definition: 

A function  $ f(x) $ is said to be periodic function, if there exists a positive real number  $ T $  , such that 

$ f\left( x+T \right)=f\left( x \right),\forall x\in R $  

Then,  $ f(x) $ is a periodic function where least positive value of T is called fundamental period. 

• Graphically: 

The function is said to be periodic if the graph repeats at a set interval, and its period is the width of that interval.

Some Standard Results on Periodic Functions:

Properties of Periodic Function:

i. If  $ f\left( x \right) $  is periodic with period  $ T $ , then

1. $ c.f\left( x \right) $  is periodic with period  $ T $  

2. $ f\left( x\pm c \right) $  is periodic with period  $ T $  

3. $ f\left( x \right)\pm c $  is periodic with period  $ T $  

Where  $ c $  is any constant

ii. If  $ f\left( x \right) $  is periodic with period   $ T $ , then

$ kf\left( cx+d \right) $  has period  $ \dfrac{T}{\left| c \right|} $  

That is Period can be only affected by coefficient of  $ x $  where \[\text{k, c, d}~\in \] constant.

iii. If   $ {{f}_{1}}\left( x \right),{{f}_{2}}\left( x \right) $  are periodic functions with periods  $ {{T}_{1}},{{T}_{2}} $  respectively, 

Then we have,

$ h\left( x \right)=a{{f}_{1}}\left( x \right)\pm b{{f}_{2}}\left( x \right) $  has period as, \[\text{LCM}\] of  $ \left\{ {{T}_{1}},{{T}_{2}} \right\} $  

Note:

1. LCM of  $ \left( \dfrac{a}{b},\dfrac{c}{d},\dfrac{e}{f} \right)=\dfrac{LCM\text{ of }(a,c,e)\text{ }}{HCF\text{ of }(b,d,f)} $  

2. \[\text{LCM}\] of rational and rational always exists. 

\[\text{LCM}\] of irrational and irrational sometime exists. 

But \[\text{LCM}\] of rational and irrational never exists.

For example, \[\text{LCM}\] of  $ \left( \text{2 }\!\!\pi\!\!\text{ , 1, 6 }\!\!\pi\!\!\text{ } \right) $  is not possible because  $ \text{2 }\!\!\pi\!\!\text{ , 6 }\!\!\pi\!\!\text{ }\in  $  irrational and  $ 1\in  $  rational.

Relations and Functions Class 11 Notes Maths - Basic Subjective Questions

Section–A (1 Mark Questions)

1. If $f(x)=x^3-\frac{1}{x^3}$, then $f(x)+f\left ( \frac{1}{x} \right )$ is equal to ______.

Ans. Since $f(x)=x^3-\frac{1}{x^3}$

$f\left(\frac{1}{x}\right)=\frac{1}{x^3}-\frac{1}{\frac{1}{x^3}}=\frac{1}{x^3}-x^3$

Hence $f(x)+f\left(\frac{1}{x}\right)=x^3-\frac{1}{x^3}+\frac{1}{x^3}-x^3=0$.

2. Find the domain of real function f defined by $f(x)=\sqrt{x-1}$.

Ans. Given that: $f(x)=\sqrt{x-1}$

$f(x)$ is defined if $x-1 \geq 0 \Rightarrow x \geq 1$

$\therefore$ Domain of $f(x)=[1, \infty)$

3. Let n (A) = m and n (B) = n, then the total number of non-empty relations that can be defined from A to B is equal to______.

Ans. Given that: $n(\mathrm{~A})=\mathrm{m}$ and $n(\mathrm{~B})=\mathrm{n}$

$\therefore n(\mathrm{~A} \times \mathrm{B})=n(\mathrm{~A}) n(\mathrm{~B})=m n$

So, the total number of relations from $A$ to $B$ $=2^{\text {min }}$.

Thus, the total number of non-empty relations from $\mathrm{A}$ to $\mathrm{B}=2^{m a}-1$.

4. The Cartesian product of A = {1, 2} and B = {3, 4} is equal to ______.

Ans. Given that: $A=\{1,2\}, B=\{3,4\}$

Therefore, $A \times B=\{(1,3),(1,4),(2,3),(2,4)\}$.

5. The function $f(x)=x+1$ and $g(x)=2x-1$. The value $\left ( \frac{f}{g} \right )(x)$ is ______.

Ans. $\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}=\frac{x+1}{2 x-1}$.

Where, $x \neq \frac{1}{2}$

Section–B (2 Marks Questions)

6. Find the Domain of the function $f(x)=log(7x+3)$.

Ans. Function $f(x)$ is defined if $7 x+3>0$

$$ \begin{aligned} & \Rightarrow 7 x+3>0 \\ & \Rightarrow 7 x>-3 \\ & \Rightarrow x>-\frac{3}{7} \end{aligned} $$

Hence, domain $\left(-\frac{3}{7}, \infty\right)$.

7. If $y=f(x)=\frac{(ax-b)}{bx-a}$, show that x=f(y).

Ans. Given:

$$ \begin{gathered} y=f(x)=\frac{(a x-b)}{(b x-a)} \\ \Rightarrow f(y)=\frac{(a y-b)}{(b y-a)} \end{gathered} $$

Let us prove that $x=f(y)$

We have, $y=\frac{(a x-b)}{(b x-a)}$

By cross-multiplying,

$$ \begin{aligned} & y(b x-a)=a x-b \\ & b x y-a y=a x-b \\ & b x y-a x=a y-b \\ & x(b y-a)=a y-b \\ & x=\frac{(a y-b)}{(b y-a)}=f(y) \\ & \therefore x=f(y) \end{aligned} $$

Hence proved.

8. Find x and y if:

(i) (4x+3,y)=(3x+5-2)

(ii) (x-y,x+y) = (6,10)

Ans. 

(i) Since $(4 x+3, y)=(3 x+5,-2)$, so

$ \begin{aligned} & 4 x+3=3 x+5 \quad \text { and } y=-2 \\ & \Rightarrow x=2 \text { and } y=-2 \end{aligned} $

(ii) $x-y=6 ; x+y=10$

$\begin{aligned} & \therefore 2 x=16 \Rightarrow x=8 \\ & \therefore y=2 . \end{aligned}$

9. If f is a real function defined by $f(x)=\frac{x-1}{x+1}$ then prove that: $f(2x)=\frac{3f(x)+1}{f(x)+3}$.

Ans. We have, $f(x)=\frac{x-1}{x+1}$

$ \Rightarrow \frac{f(x)+1}{f(x)-1}=\frac{x-1+x+1}{x-1-x-1} $

[Applying componendo and dividendo]

$ \Rightarrow x=\frac{f(x)+1}{1-f(x)} $

Now, $f(2 x)=\frac{2 x-1}{2 x+1}$

$ \begin{aligned} & \Rightarrow f(2 x)=\frac{2\left\{\frac{f(x)+1}{1-f(x)}\right\}-1}{2\left\{\frac{f(x)+1}{1-f(x)}\right\}+1} \\ & \Rightarrow f(2 x)=\frac{2 f(x)+2-1+f(x)}{2 f(x)+2+1-f(x)} \\ & \Rightarrow f(2 x)=\frac{3 f(x)+1}{f(x)+3} \end{aligned} $

Hence proved.

10. If A={1,2,3}, B {3,4} and C={4,5,6}, find.

(i) $A\times(B\cap C)$

(ii) $(A\times B)\cap (A\times C)$ 

(iii) (A\times B)\cup (A\times C) 

Ans. $A=\{1,2,3\}, B=\{3,4\}$ and $C=\{4,5,6\}$

(i) $(B \cap C)=\{4\}$

$ \begin{aligned} A \times(B \cap C) & =\{1,2,3\} \times\{4\} \\ & =\{(1,4),(2,4),(3,4)\} \end{aligned} $

\begin{aligned} & \text { (ii) }(A \times B)=\{1,2,3\} \times\{3,4\} \\ &=\{(1,3),(1,4),(2,3),(2,4),(3,3),(3,4)\} \\ &(A \times C)=\{1,2,3\} \times\{4,5,6\} \\ &=\{(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4), \\ &(3,5),(3,6)\} \\ &(A \times B) \cap(A \times C)=\{(1,4),(2,4),(3,4)\} \\\end{aligned}

11. Let $f(x)=\sqrt{x}$ and $g(x)=x$  be two functions defined in the domain $R^{+}\cup \left \{ 0 \right \}$. Find-

(i)  $(f+g)(x)$

(ii) $(f-g)(x)$

(iii) $(f.g)(x)$

(iv) $\left ( \frac{f}{f} \right )(x)$ 

Ans. Given that: $f(x)=\sqrt{x}$ and $g(x)=x$ be two functions defined in the domain $R^{+} \cup\{0\}$.

i) $(f+g)(x)=f(x)+g(x)=\sqrt{x}+x$

ii) $(f-g)(x)=f(x)-g(x)=\sqrt{x}-x$

iii) $(f \cdot g)(x)=f(x) \cdot g(x)=\sqrt{x} \cdot x=x^{3 / 2}$

iv) $\left(\frac{f}{g}\right)(x)=\frac{f(x)}{g(x)}=\frac{\sqrt{x}}{x}=\frac{1}{\sqrt{x}}$ for all $x>0$.

12. Given $R=\left \{ (x,y):x,y\epsilon W,x^{2}+y^{2}\right \}=25$ Find the domain and range of R.

Ans. We have, $R=\left \{ (x,y):x,y\epsilon W,x^{2}+y^{2}\right \}=25$

R= {(0,5), (3,4), (4,3), (5,0)}

Domain of R= Set of first element of ordered pairs in R={0,3,4,5}

Range of R= Set of second element of ordered pairs in R={5,4,3,0}.

13. If P = {x: x < 3, x$\epsilon$ N}, Q = {x: x < 2, x $\epsilon$ W}. Find        (P $\cup$Q) \times (P $\cap$ Q) where W is the set of whole numbers.

Ans. Given that: P = {x: x < 3, x$\epsilon$ N}

$\Rightarrow$ P = {1,2}

Q = {x:x<2, x $\epsilon$ W}

$\Rightarrow$ Q= {0,1}

Now, 

Now, (P$\cup$Q) = {0, 1, 2} and (P$\cap$Q) = {1}

$\therefore$ (P$\cup$Q) $\times$ (P$\cap$Q) = {(0, 1), (1, 1), (2, 1)}.

Important Formulas of Class 11 Maths Chapter 2 Relations and Functions You Shouldn’t Miss!

Here are the important formulas for Class 11 Maths Chapter 2 "Relations and Functions" that you shouldn’t miss:

1. Definition of a Relation:

A relation $ R $ from set $ A $ to set $ B $ is a subset of $ A \times B $, i.e., $ R \subseteq A \times B $.

2. Domain and Range:

• Domain: The set of all first elements of ordered pairs in the relation.

• Range: The set of all second elements of ordered pairs in the relation.

3. Types of Functions:

• One-to-One Function (Injective): $ f(x_1) = f(x_2) \implies x_1 = x_2 $.

• Onto Function (Surjective): Every element in the codomain is mapped by some element in the domain.

• One-to-One Correspondence (Bijective): The function is both one-to-one and onto.

4. Function Composition:

Composition of Functions: If $ f: A \to B $ and $ g: B \to C $, then $ (g \circ f)(x) = g(f(x)) $.

5. Inverse of a Function:

Inverse Function: If $ f: A \to B $ is a bijective function, its inverse $ f^{-1} $ satisfies $ f(f^{-1}(y)) = y $ and $ f^{-1}(f(x)) = x $.

6. Types of Functions:

• Identity Function: $ f(x) = x $.

• Constant Function: $ f(x) = c $, where $ c $ is a constant.

• Linear Function: $ f(x) = ax + b $, where $ a $ and $ b $ are constants.

7. Equivalence Relations:

A relation $ R $ on a set $ A $ is an equivalence relation if it is reflexive, symmetric, and transitive.

8. Function Notation:

• Function Value: $ f(a) $ denotes the value of function $ f $ at $ a $.

• Domain and Range Notation: $ \text{Domain}(f) $ and $ \text{Range}(f) $.

Importance of Revision Notes

• Foundational Knowledge: Class 11 Maths Chapter 2 Relations and Functions Notes PDF introduces essential concepts like types of Relations and Functions, and graphs of some particular functions which are fundamental for further study in mathematics.

• Convenient Learning: The Relations and Functions Class 11 Maths Notes PDF download offers easy access to comprehensive notes, allowing for flexible and on-the-go study.

• Effective Revision: With clear explanations and examples, the notes support efficient revision and a deeper understanding of key concepts.

• Enhanced Exam Preparation: The notes help consolidate knowledge, improving readiness for exams and boosting overall academic performance.

• Accessible Resource: The Class 11 Maths Chapter 2 Relations and Functions Notes PDF ensures that all students can benefit from quality study material without any cost.

Tips for Learning the Class 11 Maths Chapter 2 Relations and Functions

Here are some tips for learning Class 11 Maths Chapter 2 Relations and Functions:

• Understand Basic Concepts: Begin by thoroughly understanding the definitions and properties of relations and functions. Key concepts include domain, range, and types of functions like one-to-one, onto, and bijective.

• Visualise with Graphs: Use graphical representations to understand different types of functions. Plotting functions can help you visualise concepts such as domain, range, and the nature of function mappings.

• Work Through Examples: Study solved examples in your textbook and notes to see how different types of problems are approached and solved. This will give you practical insights into applying the concepts.

• Practice Problems: Regularly practise a variety of problems to strengthen your grasp of relations and functions. Focus on problems involving function composition, inverse functions, and domain/range determination.

• Review Key Definitions and Theorems: Memorise and review important definitions, theorems, and properties related to relations and functions. This will help you quickly recall them during exams.

Conclusion

Understanding the key formulas from Class 11 Maths Chapter 2 – Relations and Functions is essential for understanding the concepts and scoring well in your exams. Learn the domain and range, types of functions, function composition, and inverse functions; you'll be well-equipped to solve related problems. Regularly practising and reviewing these formulas will help enhance your knowledge and problem-solving skills. You can explore comprehensive notes and exercises available on our pages for additional resources and practise.

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