# Functions OP Malhotra S.Chand ISC Class-12 Maths Solutions Ch-2

Functions OP Malhotra S.Chand ISC Class-12 Maths Solutions Ch-2. Step by step Solutions of OP Malhotra SK Gupta Anubhuti Gangal S.Chand ISC Class-12 Mathematics with Exe-2(a), Exe-2(b), Exe-2(c), and Chapter Test Questions. Visit official Website CISCE for detail information about ISC Board Class-12 Mathematics.

## Functions OP Malhotra S.Chand ISC Class-12 Maths Solutions Ch-2

Class: | 12th |

Subject: | Mathematics |

Chapter : | Ch-2 Functions of Section -A |

Board | ISC |

Writer | OP Malhotra, SK Gupta, Anubhuti Gangal |

Publications | S.Chand Publications 2020-21 |

**-: Included Topics :- **

### Functions OP Malhotra S.Chand ISC Class-12 Maths Solutions Ch-2

**Function :**

Let X and Y be two non-empty sets. A function or mapping f from X into Y written as f : X → Y is a rule by which each element x ∈ X is associated to a unique element y ∈ Y. Then, f is said to be a function from X to Y.

The elements of X are called the domain of f and the elements of Y are called the codomain of f. The image of the element of X is called the range of X which is a subset of Y.

Note: Every function is a relation but every relation is not a function.

**Exe-2(a) **

Functions OP Malhotra S.Chand ISC Class-12 Maths Solutions Ch-2

Question 1 :

If A = {1,2,3,4} and B = {1,2,3,4,5,6} are two sets and functions f : A –> B is defined by f (x) = x + 2, ∀x ∈A, then the function f is

……………..

……………

Question 2:

Show that a function ……………………… is a bijective.

Question 3:

Let f : N ………………. one – one and into.

Question 4:

……………………..

……………………..

……………………..

Question 12:

A mapping f : N –> N , where N is the set of neutral numbers is defined as

…………………..

……………………

**Functions’s ****Types :**

**(i) Bijective or One-one and Onto Function:**

A function f : X → Y is said to be a bijective function if it is both one-one and onto.

**(ii) Composition of Functions:**

Let f : X → Y and g : Y → Z be two functions. Then, composition of functions f and g is a function from X to Z and is denoted by fog and given by (fog) (x) = f[g(x)], ∀ x ∈ X.

Note

(a) In general, fog(x) ≠ gof(x).

(b) In general, gof is one-one implies that f is one-one and gof is onto implies that g is onto.

(c) If f : X → Y, g : Y → Z and h : Z → S are functions, then ho(gof) = (hog)of.

**(iii) One-one Function or Injective Function:**

A function f : X → Y is said to be a one-one function, if the images of distinct elements of x under f are distinct, i.e. f(x_{1}) = f(x_{2} ) ⇔ x_{1} = x_{2}, ∀ x_{1}, x_{2} ∈ X

A function which is not one-one, is known as many-one function.

**(iv) Onto Function or Surjective Function:**

A function f : X → Y is said to be onto function or a surjective function, if every element of Y is image of some element of set X under f, i.e. for every y ∈ y, there exists an element X in x such that f(x) = y.

In other words, a function is called an onto function, if its range is equal to the codomain.

**(v) Invertible Function:**

A function f : X → Y is said to be invertible, if there exists a function g : Y → X such that gof = I_{x} and fog = I_{y}. The function g is called inverse of function f and is denoted by f^{-1}.

Note

(i) To prove a function invertible, one should prove that, it is both one-one or onto, i.e. bijective.

(ii) If f : X → V and g : Y → Z are two invertible functions, then gof is also invertible with (gof)^{-1} = f^{-1}og^{-1}

**Exe-2(b) **

Functions OP Malhotra S.Chand ISC Class-12 Maths Solutions Ch-2

Question 1:

If f : N –> R : f (x) = 2x – 1 / 2 and g : Q –> R : g (x)

………………….

Question 2:

(i) If f : R -> R and g : R -> R are

…………………

………………..

Question 3:

Find (gof) (3), ………………………..

………………….

………………….

Question 3:

……………………

…………………..

…………………..

Question 14:

If f (x) = ………………………………… = 2^{-1}

Question 15:

If R –> R is given by

………………….

………………….

**Transcendental Function :**

A. function, which is not algebraic, is called a transcendental function. Trigonometric, Inverse trigonometric, Exponential, Logarithmic, etc are transcendental functions.

**Explicit and Implicit Functions :**

(i) **Explicit Functions :** A function is said to be an explicit function, if it is expressed in the form y = f(x).

(ii) **Implicit Functions :** A function is said to be an implicit function, if it is expressed in the form f(x, y) = C, where C is constant.

e.g., sin (x + y) – cos (x + y) = 2

**Intervals of a Function :**

(i) The set of real numbers x, such that a ≤ x ≤ b is called a closed interval and denoted by [a, b] i.e., {x: x ∈ R, a ≤ x ≤ b}.

(ii) Set of real number x, such that a < x < b is called open interval and is denoted by (a, b)

i.e., {x: x ∈ R, a < x < b}

(iii) Intervals [a,b) = {x: x ∈ R, a ≤ x ≤ b} and (a, b] = {x: x ≠ R, a < x ≤ b} are called semi-open and semi-closed intervals.

**Exe-2(c) **

### Functions OP Malhotra S.Chand ISC Class-12 Maths Solutions Ch-2

Question 1:

(i) If f : R -> R is defined by f (x) = 2x + 3, then f^{-1} find f^{-1 }(x).

(ii) If the function f : R -> R,

defined by f (x) = 3x – 4 is insatiable,

…………………..

…………………..

Question 2:

Let the function f which is ………………………..

Question 3:

……………………….

……………………….

………………………..

Question 13:

Let f :R –> R ……………………..

Question 14:

……………………..

Question 15:

Use composition to show that f and g are inverse of each other.

………………….

……………………

**Functions :**

A function is a relationship which explains that there should be only one output for each input. It is a special kind of relation(a set of ordered pairs) which obeys a rule, i.e. every y-value should be connected to only one y-value.

Mathematically, “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 set A to set B such that the domain of f is A and no two distinct ordered pairs in f have the same first element. Also, A and B are two non-empty sets.

**Chapter Test**

### Functions OP Malhotra S.Chand ISC Class-12 Maths Solutions Ch-2

Question 1:

The function f : R –> R is defined by f(x) = 3-4x. State whether the function is one – one onto or bijective.

Question 2:

Let f : R –> R be ……………….. one nor onto

Question 3:

…………………..

…………………….

……………………..

Question 10:

If f (x) = (x+ 2 ) ……………………… (x).

-: End of Functions **OP Malhotra S. Chand **ISC Class-12 Maths Chapter-2 Solution :-

Return to :- OP Malhotra S. Chand ISC Class-12 Maths Solutions

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