I. Introduction & Basics
1. Cartesian Product ($A \times B$)
- Definition: The set of all ordered pairs $(a, b)$ where $a \in A$ and $b \in B$,.
- Cardinality: If set $A$ has $m$ elements and set $B$ has $n$ elements, then $n(A \times B) = mn$,.
2. Relations
- Definition: A relation $R$ from set $A$ to set $B$ is a subset of the Cartesian product $A \times B$,.
- Number of Relations: The total number of relations from $A$ to $B$ is $2^{n(A) \cdot n(B)}$ (equivalent to the number of subsets of $A \times B$),.
- Domain, Codomain, & Range:
- Domain: Set of inputs (first elements) that have an image,.
- Codomain: The complete set $B$,.
- Range: Set of actual outputs (second elements). Note: Range $\subseteq$ Codomain,.
II. Types of Relations (Crucial for JEE)
A relation $R$ on set $A$ is classified as:
1. Reflexive
- Condition: Every element is related to itself. $(a, a) \in R$ for all $a \in A$,.
- Formula: Total reflexive relations on a set with $n$ elements = $2^{n^2 - n}$.
2. Symmetric
- Condition: If $(a, b) \in R$, then $(b, a)$ must also belong to $R$,.
- Note: It is an "if-then" condition. If $a$ is related to $b$, the reverse must hold.
3. Transitive
- Condition: If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c)$ must belong to $R$,.
- Critical Case: If $(a, b)$ exists but no $(b, c)$ exists, the relation is transitive (vacuously true because it cannot be contradicted),.
4. Equivalence Relation
- A relation is an Equivalence Relation if it is Reflexive, Symmetric, and Transitive,.
- Standard Example: Parallel lines in a plane ($L_1 \parallel L_2$) is an equivalence relation,.
III. Functions
1. Definition
- A relation is a function if every element in the domain has a unique image in the codomain,.
- Rules:
- No element in the domain can be left empty (must have an image).
- One input cannot map to multiple outputs (Divergence is not allowed).
- Multiple inputs can map to the same output (Convergence is allowed).
2. Visual Tests
- Vertical Line Test: If a vertical line cuts the graph at more than one point, it is not a function,.
- Horizontal Line Test: If a horizontal line cuts the graph at more than one point, the function is Many-One,.
3. Classification of Functions
-
One-One (Injective): Distinct elements map to distinct images. $f(x_1) = f(x_2) \implies x_1 = x_2$,.
- Monotonicity: Strictly increasing or decreasing functions are One-One,.
-
Many-One: Two or more inputs map to the same output.
-
Onto (Surjective): Range = Codomain,.
-
Into: Range $\subset$ Codomain (at least one element in codomain is unmapped).
-
Bijective: A function that is both One-One and Onto. Only bijective functions are Invertible,.
-
Polynomial Tips:
- Odd degree polynomials (defined on $\mathbb{R}$) are always Onto (Range is $\mathbb{R}$),.
- Even degree polynomials (defined on $\mathbb{R}$) are always Many-One and Into,.
IV. Domain & Range Calculation
1. Domain Rules
- $\sqrt{f(x)} \implies f(x) \ge 0$.
- $\frac{1}{f(x)} \implies f(x) \neq 0$.
- $\log_a(b) \implies b > 0, a > 0, a \neq 1$,.
- $\sin^{-1}(x), \cos^{-1}(x) \implies x \in [-1, 1]$.
2. Range Methods
- Quadratic: Use vertex formula or complete the square.
- Linear/Linear: $y = \frac{ax+b}{cx+d}$. Range is $\mathbb{R} - {a/c}$ (ratio of leading coefficients).
- Trigonometric: Range of $a\sin x + b\cos x$ is $[-\sqrt{a^2+b^2}, \sqrt{a^2+b^2}]$,.
V. Special Functions
1. Greatest Integer Function (GIF) / Box Function $[x]$
- Returns the greatest integer less than or equal to $x$,.
- Properties:
- $[x + I] = [x] + I$ (where $I$ is an integer).
- $[x] + [-x] = 0$ if $x \in \mathbb{Z}$; otherwise $-1$,.
- Inequalities: If $[x] \ge n \implies x \in [n, \infty)$.
2. Fractional Part Function ${x}$
- Defined as ${x} = x - [x]$. Range is $[0, 1)$,.
- Periodicity: Periodic with fundamental period $T=1$,.
- Properties:
- ${x + I} = {x}$.
- ${x} + {-x} = 0$ if $x \in \mathbb{Z}$; otherwise $1$,.
3. Signum Function
- $y = \text{sgn}(x)$. Values: $1 (x>0)$, $0 (x=0)$, $-1 (x<0)$,.
VI. Operations on Functions
1. Composite Functions
- $(fog)(x) = f(g(x))$.
- Generally not commutative: $fog \neq gof$,.
- Pattern Recognition: If $f(f(x)) = x$, then $f$ is its own inverse.
2. Inverse Functions ($f^{-1}$)
- Only defined for Bijective (One-One & Onto) functions,.
- Method: Let $y = f(x) \to$ Solve for $x$ in terms of $y \to$ Swap $x$ and $y$,.
- Graph: Symmetric about the line $y = x$,.
- Property: $(gof)^{-1} = f^{-1} \circ g^{-1}$ (Reversal Law),.
VII. Functional Properties
1. Even & Odd Functions
- Even: $f(-x) = f(x)$ (Graph symmetric about Y-axis),.
- Examples: $\cos x, x^2, |x|$.
- Odd: $f(-x) = -f(x)$ (Graph symmetric about Origin),.
- Examples: $\sin x, x^3, \tan x$.
- Note: A function can be neither even nor odd (e.g., $e^x$).
2. Periodic Functions
- $f(x+T) = f(x)$ where $T$ is the least positive period,.
- Standard Periods:
- $\sin x, \cos x \to 2\pi$.
- $\tan x, |\sin x| \to \pi$.
- ${x} \to 1$.
- Scaling Property: If period of $f(x)$ is $T$, period of $f(ax)$ is $T/|a|$,.
- Sum of Functions: Period is usually LCM of individual periods (check for shorter periods if functions are complementary, e.g., $|\sin x| + |\cos x|$),.
VIII. Graphical Transformations
- Translation:
- $f(x) + a$: Shift up by $a$.
- $f(x-a)$: Shift right by $a$,.
- Reflection:
- $-f(x)$: Reflection in X-axis,.
- $f(-x)$: Reflection in Y-axis,.
- Modulus:
- $|f(x)|$: Flip negative $y$ portion to positive,.
- $f(|x|)$: Discard left of y-axis, mirror right side to left,.
- $|y| = f(x)$: Discard negative values of original graph; mirror positive graph downwards,.
IX. Functional Identities
Standard identities to memorize:
- $f(x+y) = f(x) + f(y) \implies f(x) = kx$,.
- $f(xy) = f(x) + f(y) \implies f(x) = k \ln x$,.
- $f(x+y) = f(x) \cdot f(y) \implies f(x) = a^x$,.
- $f(xy) = f(x) \cdot f(y) \implies f(x) = x^n$,.
- If polynomial satisfies $f(x)f(1/x) = f(x) + f(1/x) \implies f(x) = 1 \pm x^n$,.
Problem Solving Trick:
- For equations with $f(x)$ and $f(1/x)$, replace $x$ with $1/x$ to generate a second equation and solve simultaneously.
- For series sums involving $f(x)$, check if $f(x) + f(1-x) = \text{constant}$ (often 1) to pair terms.