MathematicsRelations and FunctionsJEE

Relations and Functions · one page

Formula sheet

Notation: R ⊆ A × B · R⁻¹ · I_A · n(A) · B_n · f : A → B · gof(x) = g(f(x)) · f⁻¹ · D = b² − 4ac · [x] · {x} = x − [x] · sgn(x) · Q · Z · W. Hollow dot: end excluded; filled dot: end included. [x] greatest integer, {x} = x − [x].

Learning path · Formula sheet · Practice set

Print and revise: Revision sheet (PDF) · Formula sheet (PDF, one page)

Part 1 · Relations: Definition, Domain and Range

n (A \times B) = pq when n (A) = p, n (B) = q

Relation as a subset of the Cartesian product

relations from A to B = 2^{pq}

Relation as a subset of the Cartesian product

Part 2 · Reflexive, Symmetric and Transitive Relations

reflexive: 2^{n^{2} - n}

Reflexive relation

symmetric: 2^{(n^{2} + n) /2}

Symmetric relation

reflexive and symmetric: 2^{(n^{2} - n) /2}

Symmetric relation

Part 3 · Equivalence Relations and Bell Numbers

B_{n + 1} = \sum_{k = 0}^{n}^{n} C_{k} B_{k}, B_{0} = 1

Equivalence relation, classes and Bell numbers

B_{1} = 1, B_{2} = 2, B_{3} = 5, B_{4} = 15

Equivalence relation, classes and Bell numbers

Part 4 · Functions: Definition, Intervals and Domain

(a, b)^{c} = (- \infty, a] \cup [b, \infty)

Intervals as subsets of R

[a, b]^{c} = (- \infty, a) \cup (b, \infty)

Intervals as subsets of R

Part 7 · Bijections, Counting Functions and Composites

functions A \to B : m^{n}

Bijective functions and counting mappings

one-one:^{m} P_{n} (m \geq n)

Bijective functions and counting mappings

onto: \sum_{r = 0}^{m} (- 1)^{r}^{m} C_{r} (m - r)^{n}

Bijective functions and counting mappings

bijections (m = n) : n!

Bijective functions and counting mappings

Part 8 · Inverse Functions

f \circ f^{- 1} = I_{B}, f^{- 1} \circ f = I_{A}

Composition and inverses

(g \circ f)^{- 1} = f^{- 1} \circ g^{- 1}

Composition and inverses

Part 9 · Inequalities and the Wavy Curve Method

x^{2} \leq a^{2} ⟺ - a \leq x \leq a

Squares and reciprocals of ranges

x^{2} \geq a^{2} ⟺ x \leq - a or x \geq a

Squares and reciprocals of ranges

ab > 0 : \frac{1}{x} \in [\frac{1}{b}, \frac{1}{a}]

Squares and reciprocals of ranges

a < 0 < b : \frac{1}{x} \in (- \infty, \frac{1}{a}] \cup [\frac{1}{b}, \infty)

Squares and reciprocals of ranges

Part 10 · Domain and Range of Algebraic Functions

f (x) = a [(x + \frac{b}{2 a})^{2} - \frac{D}{4 a ^{2}}]

Quadratic function

a > 0 : range [- \frac{D}{4 a}, \infty)

Quadratic function

a < 0 : range (- \infty, - \frac{D}{4 a}]

Quadratic function

Part 11 · Modulus Function

∣ x ∣ = x (x \geq 0), - x (x < 0)

Definition and graph

∣ f (x) ∣ = f (x) (f \geq 0), - f (x) (f < 0)

Definition and graph

∣ ab ∣ = ∣ a ∣ ∣ b ∣

Properties of modulus

\frac{a}{b} = \frac{∣ a ∣}{∣ b ∣}

Properties of modulus

x^{2} = ∣ x ∣^{2} = ∣ x^{2} ∣

Properties of modulus

∣ x ∣ = a (a > 0) \Rightarrow x = \pm a

Properties of modulus

∣ a \pm b ∣^{2} = a^{2} + b^{2} \pm 2 ab

Properties of modulus

(∣ a ∣ \pm ∣ b ∣)^{2} = a^{2} + b^{2} \pm 2∣ a ∣∣ b ∣

Properties of modulus

Part 12 · Trigonometric Functions: Graphs and Ranges

sin x \geq 0 ⟺ x \in ⋃ [2 nπ, (2 n + 1) π]

Transcendental functions and the six trig graphs

cos x \geq 0 ⟺ x \in ⋃ [2 nπ - \frac{π}{2}, 2 nπ + \frac{π}{2}]

Transcendental functions and the six trig graphs

a cos x + b sin x = a^{2} + b^{2} sin (x + α)

Range of a cos x + b sin x

range [- a^{2} + b^{2}, a^{2} + b^{2}]

Range of a cos x + b sin x

Part 14 · Inverse Trigonometric Functions

sin^{- 1} (- x) = - sin^{- 1} x