**NCERT solutions for class 11 exercise 2.3 chapter 2 Relations and Functions**

NCERT solutions for class 11 exercise 2.3 chapter 2 Relations and Functions are solved here for helping the students of 11 class maths students to clear their doubts on solving the NCERT unsolved questions of class 11 maths exercise 2.3 of chapter 2 Relations and functions. The proper understanding of exercise 2.3 is needed for every student for attempting the questions of higher class maths. All NCERT solutions of exercise 2.3 are prepared by an expert of maths who has huge experience in maths teaching. Each solution of the maths question is explained beautifully in easy language, therefore, every student can understand it.

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**NCERT solutions of class 11 maths**

Chapter 1-Sets | Chapter 9-Sequences and Series |

Chapter 2- Relations and functions | Chapter 10- Straight Lines |

Chapter 3- Trigonometry | Chapter 11-Conic Sections |

Chapter 4-Principle of mathematical induction | Chapter 12-Introduction to three Dimensional Geometry |

Chapter 5-Complex numbers | Chapter 13- Limits and Derivatives |

Chapter 6- Linear Inequalities | Chapter 14-Mathematical Reasoning |

Chapter 7- Permutations and Combinations | Chapter 15- Statistics |

Chapter 8- Binomial Theorem | Chapter 16- Probability |

**NCERT solutions of class 11 maths**

**CBSE Class 11-Question paper of maths 2015**

**CBSE Class 11 – Second unit test of maths 2021 with solutions**

**Class 11 -Physics**

**Q1.Which of the following relations are functions? Give reasons. If it is a function, determine its domain and range.**

(i) {(2, 1),(5,1),(8,1),(11,1),(14,1),(17,1)}

(ii) {2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7)}

(iii){1,3),(1,5),(2,5)}

Ans. In the ordered pair (2, 1),(5,1),(8,1),(11,1),(14,1),(17,1), all the elements 2,5,8,11,14 and 17 has have a unique image,therefore given relationship is a function.

In the set R = {(2, 1),(5,1),(8,1),(11,1),(14,1),(17,1)},the first element of the ordered pair is the domain

Therefore

Domain of the function= {2,5,8,11,14, 17}

In the set R = {(2, 1),(5,1),(8,1),(11,1),(14,1),(17,1)},the second element of the ordered pair is the range

Therefore

Range of the function= {1}

(ii) R={(2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7)}

In the given relation the elements 2,4,6,8,10,12 and 14 have a unique image,therefore given relation is a function

In the set of ordered pair {(2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7)},first element shows the domain of the function

Therefore, domain of the function = {2,4,6,8,10,12,14}

In the set of ordered pair {(2,1),(4,2),(6,3),(8,4),(10,5),(12,6),(14,7)},second element shows the range of the function

Range of the function = {1,2,3,4,5.6,7}

(iii)R={1,3),(1,5),(2,5)}

In the given relation the element 1 has two images 3 and 5, in this relation every element have not a unique image, therefore given relation is not a function

**Q2.Find the domain and range of the following real function:**

Ans. (i) The given function f(x) is defined for x ∈ R

In the value of , x can have 0, positive and negative value,so it can be expressed as follows

Since the function is defined as , therefore the function can have 0 or negative value, the function can be expressed as follows

In both cases x≥0 and x<0, f(x) have 0 and negative value

Therefore the range of the function is (-∞,0]

Since for x ∈ R, √x ≥ 0

⇒9 – x² ≥ 0

⇒9 ≥ x²

⇒x²≤ 9

**The domain is [-3,3]**

The given function is f(x) = √(9 -x²)

For any value of x within the domain [-3.3], the minimum and maximum value of f(x) decides the range of f(x)

It can be observed that for x = 0, f(x) = 3 and for x = 3, f(x) = 0

The possibel values of f(x) lies between 0 to 3

**Therefore the range of f(x) is [0,3]**

**Q3. A function f is defined by f(x) = 2x -5. Write down the values of **

**(i) f(0) (ii) f(7) (iii) f(-3)**

Ans. The given function is

f(x) = 2x -5

(i) f(0) = 2×0-5 = -5

(ii) f(7) = 2× 7- 5 = 9

(iii) f(-3) = 2×-3 -5=-11

**Q4. The function ‘t’ which maps temperature in degree Celcius into temperature in degree Fahrenheit is defined by**

**Find (i) t(0) (ii) t(28) (iii) t(-10) (iv) The value of C when t(C) =212**

Ans. The given function is

(iv) We are given t(C) =212

**C = 100**

**Q5.Find the range of each of the following functions**

**(i) f(x) = 2 -3x , x ∈ R, x > 0**

**(ii) f(x) = x²+2 , x is a real number**

**(iii)f(x) = x , x is a real number**

Ans.(i) The given function is f(x) = 2 -3x

Also given

x > 0

Multiplying by x in both sides of inequality

3x > 0

⇒2-3x < 2-0

⇒2-3x <2

⇒f(x) <2

It indicates that all possible values of f(x) are less than 2

Therefore **range of f(x) is (-∞,2)**

**(ii) f(x) = x²+2 , x is a real number**

We are given that x is a real number

Therefore

x² ≥ 0

Adding 2 both sides

x²+2 ≥ 0+2

⇒x²+2 ≥ 2

⇒f(x) ≥ 2

It indicates that all possible values of f(x) are greater and equal to 2

**Therefore** **range of f(x) is [2,∞)**

(iii) The given function is f(x) = x

Also given x is a real number

Since f(x) = x, It indicates that all possible values of f(x) are all real numbers

**Therefore range of the function is f(x) =R**

**NCERT Solutions of class 9 maths**

Chapter 1- Number System | Chapter 9-Areas of parallelogram and triangles |

Chapter 2-Polynomial | Chapter 10-Circles |

Chapter 3- Coordinate Geometry | Chapter 11-Construction |

Chapter 4- Linear equations in two variables | Chapter 12-Heron’s Formula |

Chapter 5- Introduction to Euclid’s Geometry | Chapter 13-Surface Areas and Volumes |

Chapter 6-Lines and Angles | Chapter 14-Statistics |

Chapter 7-Triangles | Chapter 15-Probability |

Chapter 8- Quadrilateral |

**NCERT Solutions of class 9 science **

**CBSE Class 9-Question paper of science 2020 with solutions**

**CBSE Class 9-Sample paper of science**

**CBSE Class 9-Unsolved question paper of science 2019**

**NCERT Solutions of class 10 maths**

**CBSE Class 10-Question paper of maths 2021 with solutions**

**CBSE Class 10-Half yearly question paper of maths 2020 with solutions**

**CBSE Class 10 -Question paper of maths 2020 with solutions**

**CBSE Class 10-Question paper of maths 2019 with solutions**

**NCERT solutions of class 10 science**

**NCERT solutions of class 10 science**

**Solutions of class 10 last years Science question papers**

**CBSE Class 10 – Question paper of science 2020 with solutions**

**CBSE class 10 -sample paper of science**

**NCERT solutions of class 12 maths**

Chapter 1-Relations and Functions | Chapter 9-Differential Equations |

Chapter 2-Inverse Trigonometric Functions | Chapter 10-Vector Algebra |

Chapter 3-Matrices | Chapter 11 – Three Dimensional Geometry |

Chapter 4-Determinants | Chapter 12-Linear Programming |

Chapter 5- Continuity and Differentiability | Chapter 13-Probability |

Chapter 6- Application of Derivation | CBSE Class 12- Question paper of maths 2021 with solutions |

Chapter 7- Integrals | |

Chapter 8-Application of Integrals |

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