**CBSE NCERT NOTES CLASS 11 MATHS CHAPTER 2 **

**Relations And Functions**

**Chapter Notes**

**Ordered Pair**

An ordered pair consists of two objects or elements in a given fixed order, e.g., (a,b).

**Equality of Ordered Pairs **

Two ordered pairs (a_{1}, b_{1}) and (a_{2}, b_{2}) are equal iff a_{1}** **= a_{2}** **and b_{1}** **=** **b_{2}.

**Cartesian Product**

Given two non-empty sets P and Q, the cartesian product P × Q is the set of all ordered pairs of elements from P and Q, i.e.,

P × Q = {(*p,q*) : *p *∈ P, *q *∈ Q }

If either P or Q is the null set, then P × Q will also be empty set, i.e.,

P × Q = φ

If there are three sets P, Q, R and p ∈ P, q ∈ Q and r ∈ R, then (p, q, r) is called an ordered triplet. Then,

P × Q × R = {(*p, q, r*) : *p *∈ P, *q *∈ Q, *r *∈ R }

**Properties of Cartesian Product**

For three sets A, B and C

- n (A × B)= n(A) n(B)
- A × B = Φ, if either A or B is Φ.
- A × (B ∪ C)= (A × B) ∪ (A × C)
- A × (B ∩ C) = (A × B) ∩ (A × C)
- A × (B - C)= (A × B) -(A × C)
- (A × B) ∩ (C × D)= (A ∩ C) × (B ∩ D)
- If A ⊆ B and C ⊆ D, then (A × C) ⊂ (B × D)
- If A ⊆ B, then A × A ⊆ (A × B) ∩ (B × A)
- A × B = B × A ⇔ A = B
- If either A or B is an infinite set, then A × B is an infinite set.
- A × (B’ ∪ C’)’ = (A × B) ∩ (A × C)
- A × (B’ ∩ C’)’ = (A × B) ∪ (A × C)
- If A and B be any two non-empty sets having n elements in common, then A × B and B × A have n
^{2}elements in common. - If A ≠ B, then A × B ≠ B × A
- If A = B, then A × B= B × A
- If A ⊆ B, then A × C ⊆ B × C

**Relation**

A relation R from a non-empty set A to a non-empty set B is a subset of the cartesian product A × B. The subset is derived by describing a relationship between the first element and the second element of the ordered pairs in A × B.

If R ⊆ A × B and (a, b) ∈ R, then we say that a is related to b by the relation R, written as aRb.

The first element is called pre-image and the second element is called the *image** *of the first element.

**Domain **

The set of all first elements of the ordered pairs in a relation R from a set A to a set B is called the *domain *of the relation R (it is not always A)

**Codomain and Range**

The set of all second elements in a relation R from a set A to a set B is called the *range *of the relation R. The whole set B is called the *codomain *of the relation R. Note that range ⊆ codomain.

Thus, domain of R = {a : (a , b) ∈ R} and range of R = {b : (a, b) ∈ R}

- A
*relation*may be represented algebraically either by the*Roster method*or by the*Set-builder method*. - An arrow diagram is a visual representation of a relation.

**Inverse Relation**

If A and B are two non-empty sets and R be a relation from A to B, such that R = {(a, b) : a ∈ A, b ∈ B}, then the inverse of R, denoted by R^{-1} , i a relation from B to A and is defined by R^{-1} = {(b, a) : (a, b) ∈ R}

**Function**

A relation *f* from a set A to a set B is said to be a function if every element of set A has one and only one image in set B.

In other words, a function *f *is a relation from a non-empty set A to a non-empty set B such that the domain of *f* is A and no two distinct ordered pairs in *f *have the same first element.

**Equal Functions**

Two functions *f* and *g* are said to be equal iff,

- Domain of
*f*= domain of*g*, - Co-domain of
*f*= co-domain of*g*and *f*(*x*)*= g*(*x*) for all*x*belonging to co-domain

**Real Valued Function and Real Function**

A function which has either R or one of its subsets as its **range** is called a ** real valued function**.

A function which has either R or one of its subsets as its **range** and **domain**, it is called a ** real function**.

**To find domain of a function**

**Case 1: **If the** **rational function is of the form $\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)}$. The denominator *g(x)* should not be zero.

**Case 2:** If the function involves square roots, then the expression within the square root should not be negative.

Solve the equations and write the answer in interval notation.

**To find range of a function**

- Write
*y = f*(*x*) - Solve
*x*in terms of*y*, let*x = g*(*y*) - Find the values of
*y*for which the values of*x*are real and in the domain of*f*. - This set is the range of
*f*(*x*)*.* *In case of complicated functions, it is better to draw graph of the function.*

**Some Real Valued Functions and Their Graphs**

**Rational functions **are functions of the type

$\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)}$,

where *f*(*x*) and *g*(*x*) are polynomial functions of *x *defined in a domain and *g*(*x*) ≠ 0.

**Congruence Modulo m**

Let m be an arbitrary but fixed integer. Two integers a and b are said to be congruence modulo m, if a – b is divisible by m and we write a ≡ b (mod m). i.e., a ≡ b (mod m) ⇔ a – b is divisible by m.

**Identity function**

A function *f *: **R **→ **R **defined** **by *y *= *f*(*x*) = *x *for each *x *∈ **R**, is called the *identity function*.

Both domain and range are **R.**

**Constant function **

The function *f *: **R **→ **R **defined** **by *y = f *(*x*) = *c*, *x *∈ **R **where *c *is a constant and each *x *∈ **R**. Here domain of *f *is **R **and its range is {*c*}.

**Polynomial function **

A function *f *: **R**→**R **is said to be *polynomial function *if for each *x *in **R**, *y *= *f *(*x*) = *a*_{0} + *a*_{1}*x *+ *a*_{2}*x*^{2} + ...+ *a*_{n}* x*^{n}*, *where *n *is a non-negative integer and *a*^{0}**, a**

^{1}

**,**

*a*^{2}

**,...,**

*a*^{n }

**∈R**.

**Examples,** *f*(*x*) = *x*^{3 }- *x*^{2 }+ 2, and *g*(*x*) = *x*^{4} + $\sqrt{2}$*x.*

But* x*^{3/2} + 2*x *is not a polynomial function. (*Why*?)

For* f*: **R **→ **R **by *y = f*(*x*) = *x*^{2}*, x *∈ **R**.

Domain of *f *= {*x *: *x *∈ **R**}.

Range of *f *= {*x *: *x *≥ 0*, x *∈ **R**}.

**Reciprocal Function**

*f *: **R **– {0} → **R **defined by *f *(*x*) = $\frac{1}{\mathrm{x}}$

*Domain = ***R **– {0}, range = **R **– {0}

**The Modulus function **

The function *f *: **R**→**R **defined by *f*(*x*) = |*x*| for each *x *∈ **R **is called *modulus function*.

$$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{c}-\mathrm{x},\mathrm{}\mathrm{}\mathrm{x}0\\ \mathrm{x},\mathrm{}\mathrm{}\mathrm{x}\ge 0\end{array}\right.$$

**Signum function **

The function *f *: **R**→**R **defined by

$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{}\left\{\begin{array}{c}1\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{i}\mathrm{f}\mathrm{}\mathrm{x}0\\ 0\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{i}\mathrm{f}\mathrm{}\mathrm{x}=0\\ -1\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{i}\mathrm{f}\mathrm{}\mathrm{x}0\end{array}\right.$

is called the *signum function.*

It can also be defined as,

$$\mathrm{f}\left(\mathrm{x}\right)=\left\{\begin{array}{c}\frac{\left|\mathrm{x}\right|}{\mathrm{x}}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{i}\mathrm{f}\mathrm{}\mathrm{x}\ne 0\\ 0\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{i}\mathrm{f}\mathrm{}\mathrm{x}=0\end{array}\right.$$

The domain of the signum function is **R **and the range is the set {–1, 0, 1}.

**Greatest integer function or floor function**

The function *f*: **R **→ **R **defined by *f*(*x*) = ⌊x⌋, *x *∈ **R, **assumes the value of the greatest integer, less than or equal to *x *is called the *greatest integer function.*

⌊x⌋= –1 for –1 ≤ *x *< 0

⌊x⌋= 0 for 0 ≤ *x *< 1

⌊x⌋= 1 for 1 ≤ *x *< 2

⌊x⌋ = 2 for 2 ≤ *x *< 3 and so on.

**Smallest integer function or ceiling function**

The function *f*: **R **→ **R **defined by *f*(*x*) = ⌈*x*⌉, *x *∈ **R, **assumes the value of the smallest integer, greater than or equal to *x *is called the *smallest integer function.*

⌈*x*⌉ = 0 for –1< *x *≤ 0

⌈*x*⌉ = 1 for 0 < *x *≤ 1

⌈*x*⌉ = 2 for 1 < *x *≤2

⌈*x*⌉ = 2 for 2 < *x *≤ 3 and so on.

**Fractional part function **

The function *f*: **R **→ **R **defined by *f*(*x*) = {*x*}, *x *∈ **R, **defined as,

{*x*} = *x* - ⌊x⌋

is called the *fractional part function.*

{1.2} = 1.2 - 1 = 0.2

{- 1.2} = -1.2 – (-2) = 0.8

The function
is called square function | |

The function,
is called square root function. |

**Exponential function**

If a is a positive real number, other than unity, then a function that maps each x ∈ R to *a*^{x} is called exponential function.

$$\mathrm{f}\left(\mathrm{x}\right)=\mathrm{}\left\{\begin{array}{c}1\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{i}\mathrm{f}\mathrm{}\mathrm{x}0\\ =0\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{i}\mathrm{f}\mathrm{}\mathrm{x}=0\\ 1\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{}\mathrm{i}\mathrm{f}\mathrm{}\mathrm{x}0\end{array}\right.$$

For 0 < a < 1, the exponential function is a strictly decreasing function

For a > 1, the exponential function is a strictly increasing function

**Logarithmic Function**

If a is a positive real number, other than unity, then,

$$\mathrm{f}\left(\mathrm{x}\right)={\mathrm{log}}_{\mathrm{b}}\mathrm{x},\mathrm{}\mathrm{x}0$$

is defined as logarithmic function. *b* is called the base.

For 0 < b < 1, the exponential function is a strictly decreasing function and is negative

For b > 1, the exponential function is a strictly increasing function and is positive.

**Laws of logarithms**

If

*x* = *b*^{ y},

then

log_{b }(*x*)* = y*

For example,

100 = 10^{2 }⇒ log_{10} (100) = 2

**Logarithm as inverse function of exponential function**

The logarithmic function,

*y *= log_{b}(*x*)

is the inverse function of the exponential function,

$$\mathrm{x}\mathrm{}=\mathrm{}{\mathrm{b}}^{\mathrm{y}}=\mathrm{}{\mathrm{b}}^{{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\mathrm{}\mathrm{x}}$$

Also *x* = log_{b }(*b*^{x})

**Natural logarithm (ln)**

*ln* (*x*) = log_{e }(*x*)

**Inverse logarithm (antilog) calculation**

The inverse logarithm (or anti logarithm) is calculated by raising the base *b* to the logarithm *y*

*x* = log^{-1}(*y*) = *b*^{ y}

**Logarithm rules**

$$\mathrm{l}\mathrm{o}\mathrm{g}\mathrm{b}\left(\mathrm{x}\mathrm{}\bullet \mathrm{}\mathrm{y}\right)=\mathrm{}{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\left(\mathrm{x}\right)+{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\left(\mathrm{y}\right)$$

$${\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\left(\frac{\mathrm{x}}{\mathrm{y}}\right)=\mathrm{}{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\left(\mathrm{x}\right)-\mathrm{}{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\left(\mathrm{y}\right)$$

$${\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\mathrm{}\left({\mathrm{x}}^{\mathrm{y}}\right)=\mathrm{}\mathrm{y}\mathrm{}\bullet \mathrm{}{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\left(\mathrm{x}\right)$$

$${\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\left(\mathrm{c}\right)=\mathrm{}\frac{1}{{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{c}}\left(\mathrm{b}\right)}$$

$${\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{b}}\left(\mathrm{x}\right)=\frac{{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{c}}\left(\mathrm{x}\right)}{{\mathrm{l}\mathrm{o}\mathrm{g}}_{\mathrm{c}}\left(\mathrm{b}\right)}$$

**Algebra of real functions**

**Addition of two real functions **

Let *f *: X → **R **and *g *: X → **R **be any two real functions, where X ⊂ **R**.

Then (*f *+ *g*): X → **R **is** **defined by

(*f *+ *g*) (*x*) = *f *(*x*) + *g *(*x*), for all *x *∈ X.

**Subtraction Real Functions **

Let *f *: X → **R **and *g *: X → **R **be any two real functions, where X ⊂ **R**.

Then (*f *+ *g*): X → **R **is** **defined by

(*f *- *g*) (*x*) = *f *(*x*) - *g *(*x*), for all *x *∈ X.

**Multiplication by a scalar **

Let *f *: X→**R **be a real valued function and k be a scalar.

Then the product k*f *is a function from X to **R **defined by

(*kf *) (*x*) = *kf *(*x*), *x *∈X.

**Multiplication of two real functions **

The product (or multiplication) of two real functions *f *: X→**R **and *g *: X→**R **is a function *f g *: X→**R **defined by

(*f g*) (*x*) = *f *(*x*) *g *(*x*), for all *x *∈ X.

This is called **pointwise multiplication**.

**Quotient of two real functions **

Let *f *and *g *be two real functions defined from X→**R **where X ⊂**R**. The quotient of *f *by *g *denoted by *f/g *is a function defined by,

$$\left(\frac{\mathrm{f}}{\mathrm{g}}\right)\left(\mathrm{x}\right)=\frac{\mathrm{f}\left(\mathrm{x}\right)}{\mathrm{g}\left(\mathrm{x}\right)},\mathrm{}\mathrm{}\mathrm{p}\mathrm{r}\mathrm{o}\mathrm{v}\mathrm{i}\mathrm{d}\mathrm{e}\mathrm{d}\mathrm{}\mathrm{g}\left(\mathrm{x}\right)\ne \mathrm{}0,\mathrm{}\mathrm{x}\mathrm{}\in \mathrm{}\mathrm{X}$$