CUET UG Mathematics Algebra Sets Relations Functions Domain Range Types of Functions

Must-Know

• A set is a well-defined collection of distinct objects; for example, A = {1, 2, 3} is a set with three elements.
• The empty set, denoted by ∅ or {}, contains no elements; {x ∈ ℝ | x² + 1 = 0} = ∅.
• A subset B of set A means every element of B is in A; {1, 2} ⊆ {1, 2, 3}.
• Power set P(A) is the set of all subsets of A; if A = {a, b}, then P(A) = {∅, {a}, {b}, {a, b}}.
• Number of elements in power set of a set with n elements is 2ⁿ; for n = 3, P(A) has 8 elements.
• Union of sets A ∪ B = {x | x ∈ A or x ∈ B}; if A = {1,2}, B = {2,3}, then A ∪ B = {1,2,3}.
• Intersection A ∩ B = {x | x ∈ A and x ∈ B}; A = {1,2}, B = {2,3} ⇒ A ∩ B = {2}.
• Difference A – B = {x | x ∈ A and x ∉ B}; A = {1,2,3}, B = {2} ⇒ A – B = {1,3}.
• Complement of set A, denoted A′ or Aᶜ, is U – A where U is universal set; if U = {1,2,3,4}, A = {1,2}, then A′ = {3,4}.
• A relation R from set A to set B is a subset of A × B; if A = {1,2}, B = {a,b}, then R = {(1,a), (2,b)} is a relation.
• A function f: A → B is a relation where every element of A has exactly one image in B; f(x) = x² from ℝ → ℝ is a function.
• Domain of a function f is the set of all input values; for f(x) = √x, domain is [0, ∞).
• Range of a function is the set of all output values; for f(x) = x², range is [0, ∞).
• Co-domain is the set B in f: A → B; it may include values not mapped by f.
• A function is one-one (injective) if f(x₁) = f(x₂) ⇒ x₁ = x₂; f(x) = 2x from ℝ → ℝ is injective.
• A function is onto (surjective) if range equals co-domain; f: ℝ → [0, ∞), f(x) = x² is surjective.
• A function is bijective if it is both injective and surjective; f: ℝ → ℝ, f(x) = 2x + 3 is bijective.
• Identity function f: A → A, f(x) = x; its domain and range are both A.
• Constant function f(x) = c for all x; domain is ℝ, range is {c}.
• Modulus function f(x) = |x| = x if x ≥ 0, –x if x < 0; domain ℝ, range [0, ∞).
• Greatest integer function f(x) = [x] is greatest integer ≤ x; [2.7] = 2, [–1.3] = –2; domain ℝ, range ℤ.
• Signum function: f(x) = 1 if x > 0, 0 if x = 0, –1 if x < 0; domain ℝ, range {–1, 0, 1}.
• A real function has domain and co-domain as subsets of real numbers; f(x) = 1/x, x ≠ 0.
• For f: A → B and g: B → C, composite function gof: A → C is defined by (gof)(x) = g(f(x)).
• If f: A → B is bijective, then inverse function f⁻¹: B → A exists such that f⁻¹(f(x)) = x.
• f(x) = sin x has domain ℝ and range [–1, 1]; it is not one-one over ℝ.
• f(x) = eˣ has domain ℝ, range (0, ∞), and is one-one.
• f(x) = logₐx (a > 0, a ≠ 1) has domain (0, ∞), range ℝ.
• (f + g)(x) = f(x) + g(x), defined where both f and g are defined.
• (fg)(x) = f(x)g(x); quotient f/g defined where g(x) ≠ 0.

Difficulty Level

Intermediate — requires understanding of definitions and ability to apply them in function analysis and set operations, but no advanced proofs.

Common CUET Traps

• Trap: Assuming all relations are functions.
  Avoid: Check that every element in domain has exactly one image; e.g., {(1,2), (1,3)} is a relation but not a function.
• Trap: Confusing range with co-domain.
  Avoid: Range is actual outputs; co-domain is target set; for f: ℝ → ℝ, f(x) = x², co-domain is ℝ but range is [0, ∞).
• Trap: Misidentifying domain of functions like 1/f(x) or √f(x).
  Avoid: For 1/f(x), exclude where f(x) = 0; for √f(x), require f(x) ≥ 0; e.g., domain of 1/√(x–1) is (1, ∞).

Practice MCQs

1. Which of the following is the domain of f(x) = √(9 – x²)?
   
   A. [–3, 3]
   
   B. (–3, 3)
   
   C. [0, 3]
   
   D. (–∞, 3]
   Answer: A
   Explanation: 9 – x² ≥ 0 ⇒ x² ≤ 9 ⇒ –3 ≤ x ≤ 3.
   Why others fail: Option B excludes endpoints where function is defined (f(±3) = 0).

2. Let A = {1, 2, 3}. How many elements are in the power set of A?
   
   A. 6
   
   B. 8
   
   C. 9
   
   D. 3
   Answer: B
   Explanation: Power set has 2ⁿ elements; n = 3 ⇒ 2³ = 8.
   Why others fail: Option A is 3! (permutation), not power set size.

3. The function f: ℝ → ℝ defined by f(x) = x³ is:
   
   A. One-one but not onto
   
   B. Onto but not one-one
   
   C. Neither one-one nor onto
   
   D. Bijective
   Answer: D
   Explanation: f(x) = x³ is strictly increasing and covers all real numbers, so both injective and surjective.
   Why others fail: Option A is common trap for f(x) = x², which is not one-one over ℝ.

4. If f(x) = |x| and g(x) = –x, then (fog)(x) is:
   
   A. x
   
   B. –x
   
   C. |–x|
   
   D. –|x|
   Answer: C
   Explanation: (fog)(x) = f(g(x)) = f(–x) = |–x|.
   Why others fail: Option A assumes |–x| = x, which fails for x < 0.

5. Let f: [0, ∞) → ℝ, f(x) = √x and g: ℝ → ℝ, g(x) = x – 1. What is the domain of (gof)?
   
   A. [0, ∞)
   
   B. [1, ∞)
   
   C. ℝ
   
   D. (–∞, 1]
   Answer: A
   Explanation: gof(x) = g(f(x)) = g(√x) = √x – 1; defined where √x is defined, i.e., x ≥ 0.
   Why others fail: Option B incorrectly assumes √x – 1 ≥ 0 is required, but domain depends only on where functions are defined.

Last-Minute Revision

• ⚠️ Empty set is subset of every set.
• ⚠️ Power set always includes ∅ and the set itself.
• ⚠️ Number of relations from A to B = 2^(m×n) if |A| = m, |B| = n.
• ⚠️ Number of functions from A to B = nᵐ if |A| = m, |B| = n.
• ⚠️ Identity function f(x) = x; symmetric about y = x.
• ⚠️ Constant function range is singleton set.
• ⚠️ Modulus function |x| is neither one-one nor onto over ℝ.
• ⚠️ Greatest integer function [x] is many-one and not onto.
• ⚠️ Signum function range: {–1, 0, 1}.
• ⚠️ f(x) = eˣ is one-one; f(x) = x² is not one-one on ℝ.
• ⚠️ logₐx defined only for x > 0 and a > 0, a ≠ 1.
• ⚠️ √f(x) requires f(x) ≥ 0.
• ⚠️ 1/f(x) requires f(x) ≠ 0.
• ⚠️ gof defined only where f(x) is in domain of g.
• ⚠️ Inverse exists only for bijective functions.
• ⚠️ [x] = n for x ∈ [n, n+1), n ∈ ℤ.
• ⚠️ |x| = √(x²) for all real x.
• ⚠️ Domain of sum/difference/product = intersection of individual domains.
• ⚠️ For f(x) = ax + b, domain and range are ℝ (if a ≠ 0).
• ⚠️ Bijective ⇔ invertible.