Relations and Functions – Class 11 Mathematics notes, important questions and MCQs

Chapter overview

Ordered pairs form Cartesian products; relations are subsets of them, and functions are special relations where each input has exactly one output.

Cartesian product and relations

• A × B = {(a, b): a ∈ A, b ∈ B}; n(A × B) = n(A) × n(B).
• A relation from A to B is any subset of A × B; number of relations = 2^(mn).
• Domain = set of first elements; range = set of second elements.

Functions

• A function assigns each element of the domain exactly one image.
• Types: identity, constant, polynomial, modulus |x|, signum, greatest integer [x].
• Algebra of functions: (f + g)(x), (fg)(x), (f/g)(x) where g(x) ≠ 0.

Key terms

Function

relation where every input has exactly one output

Range

set of all images of a function

Signum function

f(x) = 1 for x > 0, 0 for x = 0, −1 for x < 0

Formulas

• n(A × B) = n(A)·n(B)
• Number of relations = 2^(mn)

Important questions with answers

Quick revision

• (a, b) ≠ (b, a) unless a = b.
• Each input → one output.
• [x] is the greatest integer ≤ x.

Exam tips

• Vertical line test: a graph is a function if no vertical line cuts it more than once.
• For the domain of f(x) = √g(x) solve g(x) ≥ 0; for 1/g(x) remove points where g(x) = 0.

Practice MCQs

1. If n(A) = 3 and n(B) = 2, n(A × B) is

   1. 5
   2. 6
   3. 8
   4. 9
   Show answer

   Answer: (b) 6. 3 × 2.

2. [2.7] (greatest integer) equals

   1. 3
   2. 0.7
   3. 2.7
   4. 2
   Show answer

   Answer: (d) 2. Greatest integer ≤ 2.7.

3. Range of f(x) = |x| on ℝ is

   1. [0, ∞)
   2. ℝ
   3. (0, ∞)
   4. (−∞, 0]
   Show answer

   Answer: (a) [0, ∞). Non-negative.

4. Number of relations from a 2-element set to a 2-element set is

   1. 16
   2. 8
   3. 4
   4. 2
   Show answer

   Answer: (a) 16. 2⁴.

5. If f(x) = x² − 3x + 2, then f(−1) is

   1. 0
   2. 6
   3. 2
   4. −2
   Show answer

   Answer: (b) 6. 1 + 3 + 2.

6. Domain of f(x) = 1/(x − 2) is

   1. ℝ − {2}
   2. ℝ
   3. (2, ∞)
   4. ℝ − {0}
   Show answer

   Answer: (a) ℝ − {2}. Denominator ≠ 0.

7. [−2.3] equals

   1. −2
   2. 2
   3. −3
   4. −2.3
   Show answer

   Answer: (c) −3. Greatest integer ≤ −2.3.

8. Domain of f(x) = √(9 − x²) is

   1. (−3, 3)
   2. ℝ
   3. [0, 3]
   4. [−3, 3]
   Show answer

   Answer: (d) [−3, 3]. 9 − x² ≥ 0.