Relations and Functions Class 11 Notes CBSE Maths Chapter 2 [Free PDF Download]

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)}.

PDF Summary - Class 11 Maths Relations and Functions Notes (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.

Class 11 Maths Revision Notes for Chapter 2 Relations and Functions

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Cartesian Product of Sets in Relations and Functions Class 11 Explanation

A pair of elements clustered together in a specific order is called the ordered pair  Plainly, (a,b)≠(b,a)(a,b)≠(b,a).

The Cartesian product of two sets A and B is mathematically given as:

A × B = [{a, b}: a ∈ A, b ∈ B}.

Specifically, R × R = [{x, y}: x, y ∈ R)

and R × R × R = {x, y, z}: x, y, z ∈ R]

Concept of Relation in Class 11 Maths Relations and Functions Notes      

This section of class 11 math chapter 2 will include fundamentals about relations. A  relation is usually represented by letter ‘R’ from a set A to a set B is a subset of the Cartesian product A × B, which is attained by interpreting a relationship between the 1st element x and the 2nd element y of the ordered pairs in A × B, i.e., R⊆A×BR⊆A×B.

Number of Relations

Let P and Q be two non-empty finite sets, consisting of m and n elements respectively, then the total number of relations from P to Q is 2mn.

Inverse Relation

Let R ⊆ P x Q be a relation from P to Q. Then inverse of a relation R–1 ⊆ Q x P is described by R–1 = {≤ q, p): ≤ p, q) ∈ R, p ∈ P, q ∈ Q}. It is understandable now

p R q ↔ q R–1 p

Example: Let P = [1, 2, 3, 4], Q = {p, q, r} and R = {≤ 1, p), ≤ 1, r), ≤ 2, p}]. 

Then, R–1 = {≤ p, 1), ≤ r, 1), ≤ p, 2)}

Relations in a Set

Let R be a relation from P to Q. If Q = P, then R is said to be a relation in P. Hence, relation in a set P is a subset of P x P.

Identity Relation in s Set

R is said to be in identity relation if ≤ p, q ∈ R if p = q, p ∈ P, Q ∈ P. Conversely, each element of P is related to only itself.

Universal Relation in a Set

Let P be any set and R be the set P x P, then R is termed as the Universal Relation in P.

Void Relation in a Set

ϕ is known as the Void Relation in a set.

Domain In Class 11 Maths Relations and Functions Notes     

The domain of relation- R is the set of all the 1st elements of the ordered pairs in R. Domain R = {p:(p,q)∈R}{p:(p,q)∈R}.

Range In Class 11 Maths Relations and Functions Notes

The range of the relation- R is said to be the set of all 2nd elements of the ordered pairs in R. Range R = {q:(p,q)∈R}{q:(p,q)∈R}.

Function in Class 11 Maths Relations and Functions Notes   

A function represented by the letter ‘f’ from a set P to a set Q is a particular kind of relation for which every element x of set P consists of one and only one image y in set Q. Mathematically, we write f: P→Q, where f(x) = y.

Domain and Codomain in Class 11 Maths Relations and Functions Notes

A set P is known as the domain of function ‘f’ and the set Q is known as the co-domain of function ‘f’.

Image and Preimage In Class 11 Maths Relations and Functions Notes

If the element x of P corresponds to y∈Qy∈Q in the function ‘f’, then we call it as yy is the image of x under f and we mathematically write it as: f [x] =yf [x] =y. On the other hand, Pre-image is when If f [x] =yf [x] =y, then we say that xx is a preimage of y.

Kinds of Functions 

• One-to-One or Injective functions - For every element in set A, there is only one corresponding element in set B. There could be elements in set B, where there are no matching elements in set A.

• Onto or Surjective Functions - In such functions, every element of set B has one or more than one matching element in set A.

• Bijective functions are those that are one-to-one as well as Onto. This is also referred to as the perfect pairing where each member has one matching element in the other set, and no element is left out.

Important Topics Covered in this Chapter 

Some important topics covered in Relations and Functions are as follows.

• What are Relations and Functions

• Cartesian products of sets

• Range and Domain

• Representation of a relation

• Function as a special kind of relation

• Function as a correspondence

• Types of relations 

• Types of functions 

• The domain of real functions

• Some standard real functions and their graphs

• Operations on real functions

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