Relations and Functions Class 11 Notes

• Domain: set of first entries (a) in the relation.
• Codomain: the full set B from which second entries may come.
• Range: the set of second entries (b) that actually appear.
• Total number of relations from A to B with |A| = m, |B| = n is 2^(mn).

• One-one (injective): distinct inputs give distinct outputs, f(x₁) = f(x₂) implies x₁ = x₂.
• Onto (surjective): the range equals the codomain; every element of the codomain is hit.
• Bijective: both one-one and onto — it has an inverse.
• Horizontal line test: a one-one function's graph meets any horizontal line at most once.

• Identity: f(x) = x.
• Constant: f(x) = c for all x.
• Polynomial: f(x) = aₙxⁿ + ... + a₁x + a₀, defined for all real x.
• Rational: f(x) = p(x)/q(x), undefined where q(x) = 0.
• Modulus: |x| = x for x ≥ 0 and −x for x < 0.
• Signum: 1 for x > 0, 0 for x = 0, −1 for x < 0.
• Greatest integer (floor): [x] = the largest integer ≤ x.

Example: If f(x) = x² and g(x) = x + 1, find f ∘ g(3), f + g and f/g with its domain.

Solution: f ∘ g(3) = f(4) = 16. (f + g)(x) = x² + x + 1. (f/g)(x) = x²/(x + 1) with domain all real x except x = −1.

Example: Is f(x) = x² one-one on all real numbers? Is g(x) = x³ one-one?

Solution: f(x) = x² is not one-one because f(2) = f(−2) = 4. g(x) = x³ is one-one (a strictly increasing cube), meeting the horizontal line test.

Exercises (NCERT Q1–Q24)

• 1) If (x/3 + 1, y − 2/3) = (5/3, 1/3), find x and y.
• 2) If the set A has 3 elements and the set B = {3, 4, 5}, then find the number of elements in (A × B).
• 3) If G = {7, 8} and H = {5, 4, 2}, find G × H and H × G.
• 4) State whether each of the following statements is 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} and B = {3, 4}, then A × (B ∩ ∅) = ∅.
• 5) If A = {−1, 1}, find A × A × A.
• 6) 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 A × (B ∩ C) = (A × B) ∩ (A × C).
• 8) Let A = {1, 2} and B = {3, 4}. Write A × B. How many subsets will A × B have? List them.
• 9) 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, z are distinct elements.
• 10) 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.
• 11) 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.
• 12) 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 in roster form. Write down the domain and the range.
• 13) 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.
• 14) The figure shown maps the elements 5, 6, 7 of set P to 3, 4, 5 of set Q respectively. Write this relation (i) in set-builder form, (ii) in roster form. What is its domain and range?
• 15) 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.
• 16) Determine the domain and range of the relation R defined by R = {(x, x + 5) : x ∈ {0, 1, 2, 3, 4, 5}}.
• 17) Write the relation R = {(x, x³) : x is a prime number less than 10} in roster form.
• 18) Let A = {x, y, z} and B = {1, 2}. Find the number of relations from A to B.
• 19) 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.
• 20) 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)}
• 21) Find the domain and range of the following real functions: (i) f(x) = −|x|, (ii) f(x) = √(9 − x²).
• 22) A function f is defined by f(x) = 2x − 5. Write down the values of (i) f(0), (ii) f(7), (iii) f(−3).
• 23) The function t which maps temperature in degree Celsius into 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.
• 24) 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.

Answers at a glance

1. 1
   Q1 → x = 2, y = 1.
2. 2
   Q2 → A × B has 9 elements.
3. 3
   Q3 → G × H = {(7,5),(7,4),(7,2),(8,5),(8,4),(8,2)}; H × G = {(5,7),(5,8),(4,7),(4,8),(2,7),(2,8)}.
4. 4
   Q4 → (i) False → P × Q = {(m,n),(m,m),(n,n),(n,m)}. (ii) True. (iii) True.
5. 5
   Q5 → A × A × A = {(-1,-1,-1), (-1,-1,1), (-1,1,-1), (-1,1,1), (1,-1,-1), (1,-1,1), (1,1,-1), (1,1,1)}.
6. 6
   Q6 → A = {a, b}, B = {x, y}.
7. 7
   Q7 → Verified: both sides equal ∅.
8. 8
   Q8 → A × B = {(1,3),(1,4),(2,3),(2,4)}; it has 2⁴ = 16 subsets.
9. 9
   Q9 → A = {x, y, z}, B = {1, 2}.
10. 10
    Q10 → A = {−1, 0, 1}; remaining: (-1,-1), (-1,1), (0,-1), (0,0), (1,-1), (1,0), (1,1).
11. 11
    Q11 → R = {(1,3),(2,6),(3,9),(4,12)}; domain {1,2,3,4}, codomain {1,…,14}, range {3,6,9,12}.
12. 12
    Q12 → R = {(1,6),(2,7),(3,8)}; domain {1,2,3}, range {6,7,8}.
13. 13
    Q13 → R = {(1,4),(1,6),(2,9),(3,4),(3,6),(5,4),(5,6)}.
14. 14
    Q14 → R = {(x,y) : y = x − 2, x ∈ P, y ∈ Q} = {(5,3),(6,4),(7,5)}; domain {5,6,7}, range {3,4,5}.
15. 15
    Q15 → R as listed above; domain = {1,2,3,4,6}, range = {1,2,3,4,6}.
16. 16
    Q16 → Domain = {0,1,2,3,4,5}, range = {5,6,7,8,9,10}.
17. 17
    Q17 → R = {(2,8),(3,27),(5,125),(7,343)}.
18. 18
    Q18 → 2⁶ = 64 relations.
19. 19
    Q19 → Domain = Z, range = Z.
20. 20
    Q20 → (i) and (ii) are functions; (iii) is not a function.
21. 21
    Q21 → (i) Domain R, range (−∞, 0]. (ii) Domain [−3, 3], range [0, 3].
22. 22
    Q22 → f(0) = −5, f(7) = 9, f(−3) = −11.
23. 23
    Q23 → t(0) = 32, t(28) = 82.4, t(−10) = 14, and C = 100 when t(C) = 212.
24. 24
    Q24 → (i) (−∞, 2). (ii) [2, ∞). (iii) R.

MCQ Questions

1 Mark Questions

1. 1
   Define the domain of a relation.
2. 2
   Write the range of f(x) = x² for x ∈ R. (Ans: [0, ∞))
3. 3
   For f(x) = 3x − 2, find f(0). (Ans: −2)
4. 4
   State the condition for a relation to be a function. (Ans: each x maps to exactly one y)

2 Mark Questions

1. 1
   If f(x) = x³, find f(1), f(−1) and f(2). (Ans: 1, −1, 8)
2. 2
   Find the domain and range of f(x) = 1/(x − 1). (Ans: domain R − {1}, range R − {0})
3. 3
   Express the relation {(1, 2), (2, 4), (3, 6), (4, 8)} as a rule in set-builder form. (Ans: {(x, 2x) : x ∈ {1, 2, 3, 4}})

3 Mark Questions

1. 1
   Let R = {(x, y) : x, y ∈ N and x − 3y = 0}. Write R in roster form and find its domain and range. (Ans: R = {(3, 1), (6, 2), (9, 3), …}; domain {3, 6, 9, …}, range {1, 2, 3, …})
2. 2
   If f(x) = x² + 2 and g(x) = 2x − 1, find (f∘g)(x) and (g∘f)(x). (Ans: (2x − 1)² + 2; 2x² + 3)
3. 3
   Define the greatest integer function and evaluate f(2.7) and f(−2.7). (Ans: 2, −3)

4 Mark Questions

1. 1
   Let f(x) = √(x − 1) and g(x) = 1/(x + 1). Find the domains of f and g, then of f∘g and g∘f, stating each set explicitly and justifying. (Ans: f∘g domain x ≥ ... after solving; g∘f domain x > 1)
2. 2
   Draw the graph of the absolute-value function f(x) = |x| for x ∈ [−3, 3], state its domain, range and the points where it is not differentiable, and write it as a piecewise function. (Ans: piecewise x for x ≥ 0, −x for x < 0; not differentiable at 0)