Functions & Relations

Numbers & Their Sets

• Natural Numbers (N) = {1, 2, 3, 4, …}
• Whole Numbers (W) = {0, 1, 2, 3, …}
• Integers (Z) = {…, −3, −2, −1, 0, 1, 2, 3, …}
• Rational Numbers (Q): Numbers of the form p/q, where p, q ∈ Z and q ≠ 0
• Irrational Numbers: Cannot be written as p/q (e.g. √2, √5, π, e)
• Real Numbers (R): Rational + Irrational numbers
• Positive Reals: R⁺ = (0, ∞)
• Negative Reals: R⁻ = (−∞, 0)
• R₀ = R − {0}
• Imaginary Numbers: √(−k), where k ∈ R

Intervals

• Closed Interval: [a, b] = {x : a ≤ x ≤ b}
• Open Interval: (a, b) = {x : a < x < b}
• Left Closed: [a, b) = {x : a ≤ x < b}
• Right Closed: (a, b] = {x : a < x ≤ b}

Function

• A function f : A → B assigns each element of A to a unique element of B.
• Notation: f(a) = b, where a ∈ A and b ∈ B
• Domain: Set A
• Co-domain: Set B
• Range: {f(a) : a ∈ A} ⊆ B

Testing Whether a Relation Is a Function

• Every element of domain must have an image
• No element of domain should have more than one image

Function as a Set of Ordered Pairs

• f ⊆ A × B
• No two ordered pairs have the same first element
• Every element of A appears exactly once as a first element

Special Functions

Identity Function

• f(x) = x
• Domain = Range = R

Constant Function

• f(x) = c (constant)
• Range contains only one value

Modulus Function

• |x| = x, x ≥ 0
• |x| = −x, x < 0

Signum Function

• sgn(x) = 1, x > 0
• sgn(x) = 0, x = 0
• sgn(x) = −1, x < 0

Greatest Integer Function

• [x] = greatest integer ≤ x
• [x] ≤ x < [x] + 1
• [x + n] = [x] + n, n ∈ Z

Fractional Part Function

• {x} = x − [x]
• Range = [0, 1)

Trigonometric Functions (Key Points)

• sin x, cos x: Domain = R, Range = [−1, 1]
• tan x: Domain = R − {(2n+1)π/2}, Range = R
• sec x, cosec x: Range = R − (−1, 1)
• cot x: Domain = R − {nπ}, Range = R

Inverse Trigonometric Functions (Ranges)

• sin⁻¹x ∈ [−π/2, π/2]
• cos⁻¹x ∈ [0, π]
• tan⁻¹x ∈ (−π/2, π/2)
• cot⁻¹x ∈ (0, π)

Even & Odd Functions

• Even: f(−x) = f(x) (symmetry about y-axis)
• Odd: f(−x) = −f(x) (symmetry about origin)
• Any function can be written as sum of an even and odd function

Monotonic Functions

• Increasing: x₁ < x₂ ⇒ f(x₁) ≤ f(x₂)
• Decreasing: x₁ < x₂ ⇒ f(x₁) ≥ f(x₂)
• Strict versions replace ≤ / ≥ with < / >

Periodic Function

• f(x + T) = f(x)
• Smallest such T is the fundamental period
• Example: sin x has period 2π

Types of Functions

• One-One (Injective): f(x₁) = f(x₂) ⇒ x₁ = x₂
• Many-One: Different x give same image
• Onto (Surjective): Range = Co-domain
• Into: Range ≠ Co-domain
• Bijection: One-one and onto

Composite Function

• (g ∘ f)(x) = g(f(x))
• Defined only if Range of f ⊆ Domain of g
• Not commutative, but associative

Inverse Function

• Exists only for bijections
• (f⁻¹)⁻¹ = f
• (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹

Relations

• Relation ⊆ A × B
• Total relations from A to B = 2^mn

Types of Relations

• Reflexive
• Symmetric
• Transitive
• Equivalence (all three)
• Anti-symmetric