Functions & Graphs: JEE Main Complete Guide

Functions are the language of mathematics — every other JEE Main topic uses function notation, and errors in domain/range analysis cascade into errors throughout calculus, coordinate geometry, and trigonometry. As a direct topic, functions contribute one to two questions per session. As a foundation, understanding functions correctly prevents systematic errors across the entire paper. This guide covers every JEE Main concept in the functions chapter.

Domain and Range

Domain: the set of all input values for which the function is defined. Range: the set of all output values. Finding domain requires identifying restrictions: (1) denominators cannot be zero; (2) arguments of even roots must be ≥ 0; (3) arguments of logarithms must be > 0; (4) arguments of arcsin and arccos must be in [−1, 1]; (5) tan x is undefined at x = π/2 + nπ. For composite restrictions, apply all conditions simultaneously (intersect the domains).

Finding range: (1) for simple functions, set y = f(x) and solve for x in terms of y — the domain of x gives the range of y; (2) for quadratic functions: range is [vertex y-value, ∞) or (−∞, vertex y-value] depending on opening direction; (3) graphical approach — sketch the function and read off the y-values covered. JEE Main tests domain and range of composite functions like f(g(x)) — find the range of g first, then intersect with the domain of f. For the calculus connection, see our Continuity & Differentiability Guide.

Types of Functions

Injective (one-to-one): each element of the codomain is mapped to by at most one element of the domain. Horizontal line test: if every horizontal line intersects the graph at most once, the function is injective. Surjective (onto): every element of the codomain has at least one pre-image. Range equals codomain. Bijective: both injective and surjective — has an inverse function. JEE Main tests: is a given function injective? surjective? onto a given codomain? — apply the definitions systematically. Take a free mock test on functions to practise domain, range, and injectivity questions.

Composite and Inverse Functions

Composite function: (f∘g)(x) = f(g(x)). Domain of f∘g: all x in domain of g such that g(x) is in domain of f. JEE Main tests f∘g ≠ g∘f — composition is not commutative in general. Inverse function f⁻¹ exists if and only if f is bijective. If y = f(x), then x = f⁻¹(y). Method: set y = f(x), solve for x in terms of y, then swap x and y. Domain of f⁻¹ = range of f; range of f⁻¹ = domain of f. Key property: (f∘f⁻¹)(x) = x and (f⁻¹∘f)(x) = x.

Graphs: f⁻¹ is the reflection of f in the line y = x. If f is increasing, f⁻¹ is also increasing. If f is decreasing, f⁻¹ is also decreasing.

Graph Transformations

Systematic transformations: y = f(x+a) shifts graph left by a (if a > 0). y = f(x−a) shifts right. y = f(x) + a shifts up. y = f(x) − a shifts down. y = f(−x) reflects in y-axis. y = −f(x) reflects in x-axis. y = f(ax) compresses x-axis by factor a. y = af(x) stretches y-axis by factor a. Combinations: y = |f(x)| — reflect any part of the graph below the x-axis to above it. y = f(|x|) — take the right half (x ≥ 0) and reflect it in the y-axis.

Even, Odd, and Periodic Functions

Even function: f(−x) = f(x) — symmetric about y-axis. Examples: x², cos x, |x|. Odd function: f(−x) = −f(x) — symmetric about origin. Examples: x³, sin x, x. Periodic function: f(x+T) = f(x) for some T > 0. Period of sin(ax+b) = 2π/|a|. JEE Main tests: is a given function even or odd? What is the period of |sin x|? (Period of |sin x| = π, half the period of sin x.) For trigonometric function analysis, see our Trigonometry Guide.

Functional Equations

JEE Main occasionally presents a functional equation: f(x+y) = f(x) + f(y) (Cauchy's equation, implies f(x) = cx for continuous functions). f(xy) = f(x) + f(y) (implies f(x) = c·log x). f(xy) = f(x)f(y) (implies f(x) = x^c). JEE Main tests these standard functional equations by asking you to find f(2), f(0), or f(f(x)) given a functional equation and one boundary condition.

Exam Strategy

Functions questions in JEE Main reward systematic approach: for domain, list all restrictions and intersect; for range, use the solve-for-x method; for injectivity, use the horizontal line test or show f(x₁) = f(x₂) → x₁ = x₂. Practise five questions of each type — domain, range, inverse, composition, graph transformation — until the approach is automatic. For the sets foundation, see our Sets, Relations and Functions Guide. Upgrade for ₹149/month for 200+ function problems with complete domain and range analysis.