Function and Its Properties JEE Paper 2A 2027

You may find it difficult to distinguish between the domain, range and codomain of a function or identify whether a function is one-one, many-one, onto or into. These concepts require more than memorising definitions, as solving JEE Paper 2A questions often involves applying function properties to equations, mappings and graphs.

PW's Design and Architecture programmes support your preparation with live and recorded classes, Daily Practice Problems (DPPs), quizzes and mock tests. These learning resources can complement the study of calculus basics and help students practise mathematical concepts relevant to design and architecture entrance examinations.

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Domain, Range and Codomain: Core Concepts of Functions

A function maps each input from a set to exactly one output. The domain, range and codomain describe the inputs and outputs involved in this mapping.

• Domain: The set of all permitted input values for a function.

• Codomain: The target set in which the function's outputs are defined.

• Range: The set of output values the function actually produces.

Formula:

f: A → B

Here, A is the domain and B is the codomain. The range is the subset of B containing all actual output values.

Example:

f(x) = x²

f: ℝ → ℝ

Domain = ℝ

Codomain = ℝ

Range = [0, ∞)

The square of a real number cannot be negative, so the range includes zero and all positive real numbers.

Function Criteria: Unique Output and Mapping Rules

A relation qualifies as a function when every input in its domain is associated with exactly one output. Different inputs may produce the same output, but a single input cannot have two different outputs.

For example, f(x) = x² is a function because each real number has one square. However, f(2) = 4 and f(-2) = 4, so it is not one-one over the real numbers.

The basic function rule can be expressed as:

For every x ∈ A, there is exactly one f(x) ∈ B.

Types of Functions: One-One, Many-One, Onto and Into

Functions can be classified according to how inputs map to outputs and whether every element of the codomain is reached.

A function may belong to more than one category. For example, a function can be both one-one and into, or both one-one and onto.

Injective Function: One-One Function Explained

An injective function, also called a one-one function, maps distinct inputs to distinct outputs.

Formula:

f(a) = f(b) ⇒ a = b

This means that if two outputs are equal, their corresponding inputs must also be equal.

Example:

f(x) = 2x

For real inputs, if f(a) = f(b), then 2a = 2b, which gives a = b. Therefore, this function is one-one.

To identify an injective function, check whether two different inputs can produce the same output. If they can, the function is not injective.

Many-One Function: Concept and Examples

A many-one function maps two or more different inputs to the same output. It still satisfies the basic definition of a function because each input has exactly one output.

Example:

f(x) = x²

Domain = ℝ

Codomain = ℝ

f(2) = 4

f(-2) = 4

Since two distinct inputs produce the same output, the function is many-one over the real numbers.

Many-one mappings are common in quadratic functions and other functions where different inputs can produce identical values.

Surjective Function: Onto Function Explained

A surjective function, also called an onto function, reaches every element of its codomain. Its range and codomain are therefore equal.

Formula:

Range = Codomain

An equivalent condition is:

For every y ∈ B, there exists x ∈ A such that f(x) = y.

Example:

f(x) = x³

f: ℝ → ℝ

Every real number has a real cube root. Hence, every real number in the codomain is reached, making the function onto.

When checking surjectivity, compare the range with the stated codomain. The formula alone does not determine whether a function is onto.

Into Function: Concept and Examples

An into function does not reach every element of its codomain. Its range is a proper subset of the codomain.

Formula:

Range ⊊ Codomain

The symbol ⊊ indicates a proper subset.

Example:

f(x) = x²

f: ℝ → ℝ

Range = [0, ∞)

Codomain = ℝ

Negative real numbers are part of the codomain but are never outputs of this function. Therefore, it is into and many-one.

Remember that whether a function is onto or into depends on the codomain specified in the question, not just on its formula.

Function Classification: Linear, Quadratic and Exponential Functions

Functions can also be classified by their algebraic form. Recognising these forms helps you understand their behaviour and interpret their graphs.

Formula:

f(x) = mx + c

Here, m and c are constants. The graph is a straight line. If m ≠ 0, the function is one-one over the real numbers.

Quadratic Function

Formula:

f(x) = ax² + bx + c, where a ≠ 0

Here, a, b and c are constants. The graph is a parabola. Over the real numbers, a quadratic function is generally many-one because inputs on opposite sides of its vertex may produce the same output.

Exponential Function

Formula:

f(x) = aˣ, where a > 0 and a ≠ 1

The graph represents exponential growth when a > 1 and exponential decay when 0 < a < 1. For real inputs and positive bases other than one, the function is one-one and has a range of (0, ∞).

The domain and codomain affect whether a function in any of these categories is one-one, onto or into. For instance, a quadratic function over all real numbers is generally many-one, while a linear function with a non-zero slope is one-one over the real numbers.

Horizontal Line Test: Graphical Method for Function Classification

The horizontal line test helps determine whether a function is one-one by examining its graph.

• If every horizontal line intersects the graph at most once, the function is one-one.

• If any horizontal line intersects the graph more than once, the function is not one-one.

For example:

f(x) = x³

This function passes the horizontal line test and is one-one over the real numbers.

f(x) = x²

This function fails the horizontal line test because some horizontal lines intersect its graph twice. It is therefore not one-one over the real numbers.

The horizontal line test identifies whether a function is one-one, but it does not establish whether it is onto. To determine that, you must compare its range with its codomain.

Bijective Function: Understanding One-One and Onto Functions

A bijective function is both injective and surjective. Each element of the codomain is reached by exactly one element of the domain.

Formula:

Bijective = Injective + Surjective

Example:

f(x) = x³

f: ℝ → ℝ

This function is one-one and onto, so it is bijective.

A bijective function has an inverse function because each output corresponds to exactly one input.

Periodic Functions: Concept and Properties

A periodic function repeats its values at regular intervals. A function f(x) is periodic if there is a positive constant T such that the following condition holds for every x in its domain:

Formula:

f(x + T) = f(x), where T > 0

The smallest positive value of T, when it exists, is called the fundamental period.

Examples:

f(x) = sin x

Fundamental period = 2π

f(x) = cos x

Fundamental period = 2π

Over the real numbers, both sine and cosine are many-one functions because different inputs can produce the same output. Their range is [-1, 1].

When solving questions on periodic functions, focus on the period, range and repeated values. These properties also help you interpret their graphs.

Prepare for JEE Paper 2A with Physics Wallah

Preparing for JEE Paper 2A requires a clear understanding of mathematical concepts, regular problem-solving practice and familiarity with the exam pattern. PW's Design and Architecture programmes support you through live and recorded classes, practice questions, quizzes and mock tests covering subjects such as Mathematics, Design Aptitude, Reasoning and Fine Arts.

You should use the following resources alongside independent study:

• Online paid batches: Live classes, recorded lessons, Daily Practice Problems (DPPs), doubt-solving sessions and faculty mentorship.

• Free learning resources: Selected concept videos, introductory lessons and daily quizzes to practise aptitude and strengthen fundamentals.

• Mock tests and practice exercises: Opportunities to revise concepts, assess preparation and become familiar with exam-style questions.

• Faculty support: Educators with backgrounds in Mathematics, Reasoning, Fine Arts and Design to support subject learning and problem-solving.

Functions and their properties form an important part of mathematical understanding for JEE Paper 2A 2027. Regular question-solving can help you apply these concepts accurately. PW's Design and Architecture learning resources, including classes, quizzes, DPPs and mock tests, can complement your preparation and revision.