Important Formulas: Functions | Quantitative Aptitude (Quant) - CAT PDF Download

Functions describe a relationship between inputs and outputs. Understanding functions is essential for solving problems on composition, inverses, transformations, and algebraic manipulations.

Definition of a Function

A function f from a set X to a set Y is a rule that assigns exactly one element y ∈ Y to each x ∈ X, denoted as f(x) = y.

• Domain: Set of all possible input values (x).

• Range: Set of all possible output values (f(x)).

Types of Functions

• One-to-One (Injective): Each element of the domain maps to a unique element of the range.

• Onto (Surjective): Every element of the range has a pre-image in the domain.

• Bijective: Function that is both one-to-one and onto; only bijective functions have inverses.

Standard Function Forms

Linear Function: f(x) = mx + c

• Domain: ℝ
• Range: ℝ
• Graph: Straight line, slope m, intercept c

Note: The symbol ℝ represents the set of all real numbers (positive, negative, and zero, including fractions and decimals).

Quadratic Function: f(x) = ax² + bx + c, a ≠ 0

• Domain: ℝ

• Range:

  1. a > 0 → [minimum at vertex, ∞)

  2. a < 0 → (−∞, maximum at vertex]

• Vertex: x = −b/(2a)

• Value at vertex: f(−b/2a) = −Δ/(4a), where Δ = b² − 4ac

Absolute Value Function: f(x) = |x|

• Domain: ℝ
• Range: [0, ∞)
• Property: f(x) ≥ 0

Identity Function: f(x) = x

• Domain: ℝ
• Range: ℝ

Constant Function: f(x) = c

• Domain: ℝ
• Range: {c}

Function Operations

• Sum: (f + g)(x) = f(x) + g(x)
• Difference: (f − g)(x) = f(x) − g(x)
• Product: (f × g)(x) = f(x) × g(x)
• Quotient: (f ÷ g)(x) = f(x) ÷ g(x), g(x) ≠ 0

Composition of Functions

(f ∘ g)(x) = f(g(x))

• Domain: All x ∈ Domain(g) such that g(x) ∈ Domain(f)
• Example: f(x) = x², g(x) = x + 1 → (f ∘ g)(x) = f(g(x)) = (x + 1)²

Property: (f ∘ g) ≠ (g ∘ f) in general

Inverse Functions

• f⁻¹(x) reverses the mapping of f(x).

• Condition: f must be bijective to have an inverse.

• Property: f(f⁻¹(x)) = f⁻¹(f(x)) = x

Finding an inverse:

1. Write y = f(x)

2. Solve for x in terms of y

3. Swap x and y → x = f⁻¹(y)

Example:
f(x) = 2x + 3 → y = 2x + 3 → x = (y − 3)/2 → f⁻¹(x) = (x − 3)/2

Special Properties

• Even Function: f(−x) = f(x) → symmetric about the y-axis

• Odd Function: f(−x) = −f(x) → symmetric about the origin

• Periodic Function: f(x + p) = f(x) → p is the period

Tricks for CAT

• To simplify composite functions: substitute the inner function first.
• For inverses: always verify f⁻¹(f(x)) = x.
• For quadratic and absolute value functions: check vertex and domain restrictions.
• Recognise even/odd/periodic properties to avoid unnecessary calculation.