Class 11 Mathematics

Chapter 2 Solutions — Relations and Functions

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Solutions

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Step-by-step NCERT solutions for Relations and Functions (Chapter 2, NCERT Class 11 Mathematics) — every question and answer worked out in full, not just the final result. You can also read the Relations and Functions textbook chapter.

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All 36 questions in Relations and Functions are solved in the PDF. Here's what's inside, exercise by exercise:

Exercise 2.1

If (x/3 + 1, y - 2/3) = (5/3, 1/3), find the values of x and y.
If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A × B).
If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.
4.
State whether each of the following statements are true or false. If the statement is false, rewrite the given statement correctly.
- (i) If P = {m, n} and Q = {n, m}, then P × Q = {(m, n),(n, m)}.
- (ii) If A and B are non-empty sets, then A × B is a non-empty set of ordered pairs (x, y) such that x ∈ A and y ∈ B.
- (iii) If A = {1, 2}, B = {3, 4}, then A × (B ∩ φ) = φ.
If A = {–1, 1}, find A × A × A.
If A × B = {(a, x),(a, y), (b, x), (b, y)}. Find A and B.
7.
Let A = {1, 2}, B = {1, 2, 3, 4}, C = {5, 6} and D = {5, 6, 7, 8}. Verify that
- (i) A × (B ∩ C) = (A × B) ∩ (A × C)
- (ii) A × C is a subset of B × D.
Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.
Let A and B be two sets such that n(A) = 3 and n(B) = 2. If (x, 1), (y, 2), (z, 1) are in A × B, find A and B, where x, y and z are distinct elements.
The Cartesian product A × A has 9 elements among which are found (–1, 0) and (0, 1). Find the set A and the remaining elements of A × A.

Exercise 2.2

Let A = {1, 2, 3,...,14}. Define a relation R from A to A by R = {(x, y) : 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range.
Define a relation R on the set N of natural numbers by R = {(x, y) : y = x + 5, x is a natural number less than 4; x, y ∈ N}. Depict this relationship using roster form. Write down the domain and the range.
A = {1, 2, 3, 5} and B = {4, 6, 9}. Define a relation R from A to B by R = {(x, y): the difference between x and y is odd; x ∈ A, y ∈ B}. Write R in roster form.
4.
The Fig 2.7 shows a relationship between the sets P and Q. Write this relation
- (i) in set-builder form
- (ii) roster form. What is its domain and range?
5.
Let A = {1, 2, 3, 4, 6}. Let R be the relation on A defined by {(a, b): a, b ∈ A, b is exactly divisible by a}.
- (i) Write R in roster form
- (ii) Find the domain of R
- (iii) Find the range of R.
Determine the domain and range of the relation R defined by R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.
Write the relation R = {(x, x³) : x is a prime number less than 10} in roster form.
Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.
Let R be the relation on Z defined by R = {(a, b): a, b ∈ Z, a – b is an integer}. Find the domain and range of R.

Exercise 2.3

1.
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)}
2.
Find the domain and range of the following real functions:
- (i) f(x) = –x
- (ii) f(x) = √(9 – x²)
3.
A function f is defined by f(x) = 2x – 5. Write down the values of
- (i) f(0),
- (ii) f(7),
- (iii) f(–3).
4.
The function 't' which maps temperature in degree Celsius into temperature in degree Fahrenheit is defined by t(C) = 9C/5 + 32. Find
- (i) t(0)
- (ii) t(28)
- (iii) t(–10)
- (iv) The value of C, when t(C) = 212.
5.
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.

Miscellaneous Exercise

The relation f is defined by f(x) = x², 0 ≤ x ≤ 3; f(x) = 3x, 3 ≤ x ≤ 10. The relation g is defined by g(x) = x², 0 ≤ x ≤ 2; g(x) = 3x, 2 ≤ x ≤ 10. Show that f is a function and g is not a function.
If f(x) = x², find (f(1.1) – f(1))/(1.1 – 1).
Find the domain of the function f(x) = (x² + 2x + 1)/(x² – 8x + 12).
Find the domain and the range of the real function f defined by f(x) = √(x – 1).
Find the domain and the range of the real function f defined by f(x) = |x – 1|.
Let f = {(x, x²/(1 + x²)) : x ∈ R} be a function from R into R. Determine the range of f.
Let f, g : R→R be defined, respectively by f(x) = x + 1, g(x) = 2x – 3. Find f + g, f – g and f/g.
Let f = {(1,1),(2,3),(0,–1),(–1,–3)} be a function from Z to Z defined by f(x) = ax + b, for some integers a, b. Determine a, b.
9.
Let R be a relation from N to N defined by R = {(a, b) : a, b ∈ N and a = b²}. Are the following true?
- (i) (a, a) ∈ R, for all a ∈ N
- (ii) (a, b) ∈ R, implies (b, a) ∈ R
- (iii) (a, b) ∈ R, (b, c) ∈ R implies (a, c) ∈ R. Justify your answer in each case.
10.
Let A = {1,2,3,4}, B = {1,5,9,11,15,16} and f = {(1,5),(2,9),(3,1),(4,5),(2,11)}. Are the following true?
- (i) f is a relation from A to B
- (ii) f is a function from A to B. Justify your answer in each case.
Let f be the subset of Z × Z defined by f = {(ab, a + b) : a, b ∈ Z}. Is f a function from Z to Z? Justify your answer.
Let A = {9, 10, 11, 12, 13} and let f : A→N be defined by f(n) = the highest prime factor of n. Find the range of f.

Sample

Worked examples

3 questions from Relations and Functions, solved step by step.

Exercise 2.1

If (x/3 + 1, y - 2/3) = (5/3, 1/3), find the values of x and y.

Since the two ordered pairs are equal, their corresponding elements must be equal.

From the first components:

x/3 + 1 = 5/3

x/3 = 5/3 - 1 = 5/3 - 3/3 = 2/3

x = 2

From the second components:

y - 2/3 = 1/3

y = 1/3 + 2/3 = 3/3 = 1

Answer: x = 2, y = 1

Exercise 2.2

Let A = {1, 2, 3,...,14}. Define a relation R from A to A by R = {(x, y) : 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range.

We need 3x - y = 0, i.e., y = 3x, with both x and y in A = {1,2,...,14}.

Finding valid pairs:

x = 1 ⇒ y = 3. (1,3) ∈ R

x = 2 ⇒ y = 6. (2,6) ∈ R

x = 3 ⇒ y = 9. (3,9) ∈ R

x = 4 ⇒ y = 12. (4,12) ∈ R

x = 5 ⇒ y = 15. 15 ∉ A, so stop here.

R = {(1,3),(2,6),(3,9),(4,12)}

Domain = set of first elements = {1, 2, 3, 4}

Codomain = A = {1, 2, 3, ..., 14}

Range = set of second elements = {3, 6, 9, 12}

Answer: R = {(1,3),(2,6),(3,9),(4,12)}; Domain = {1,2,3,4}; Codomain = {1,2,...,14}; Range = {3,6,9,12}

Exercise 2.3

Find the domain and range of the following real functions: (i) f(x) = –x (ii) f(x) = √(9 – x²)

(i) f(x) = -x

f(x) is defined for all real numbers x.

Domain = R.

For every real number x, -x is also real. As x ranges over all of R, -x also takes all real values.

Range = R.

(ii) f(x) = √(9 - x²)

For f(x) to be defined (real-valued), we need 9 - x² ≥ 0.

9 - x² ≥ 0 ⇒ x² ≤ 9 ⇒ -3 ≤ x ≤ 3.

Domain = [-3, 3].

When x ∈ [-3, 3]: x² ranges from 0 to 9, so 9 - x² ranges from 0 to 9.

Therefore √(9 - x²) ranges from 0 to 3.

Range = [0, 3].

Answer: (i) Domain = R, Range = R. (ii) Domain = [–3, 3], Range = [0, 3].

All 36 questions in Relations and Functions are solved step by step in the Solutions PDF above.

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Chapter 2 Solutions — Relations and Functions

Overview

What these solutions cover

Exercise 2.1

Exercise 2.2

Exercise 2.3

Miscellaneous Exercise

Worked examples

If (x/3 + 1, y - 2/3) = (5/3, 1/3), find the values of x and y.

Let A = {1, 2, 3,...,14}. Define a relation R from A to A by R = {(x, y) : 3x – y = 0, where x, y ∈ A}. Write down its domain, codomain and range.

Find the domain and range of the following real functions: (i) f(x) = –x (ii) f(x) = √(9 – x²)

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