Functions in Class 11 Commerce Maths – Types, Graphs & Inverse Functions Explained

After building your foundation with sets and relations, the next step is functions — one of the most important and consistently tested concepts across every commerce entrance exam in India. CUET, IPMAT, NPAT, and SET all feature function-related questions. This guide gives you a complete, exam-focused walkthrough of Chapter 2 of the Maharashtra State Board Class 11 Commerce Maths syllabus.

📊 Interactive Practice: Visualise function operations and inverses with our Interactive Function Invertibility & Composite Sandbox in the middle of this guide!

What Is a Function? The Core Definition

A function f from set A to set B is a relation that associates each element of A with exactly one element of B. This "exactly one" rule is what separates a function from a general relation.

• Domain: Set A — the inputs
• Co-domain: Set B — the target set
• Range: The actual set of all outputs — always a subset of the co-domain

If x ∈ A and y is its corresponding element in B, we write y = f(x), and say y is the image of x under f.

The golden rule: If any single element in A maps to more than one element in B, the relation is NOT a function. But multiple elements in A can map to the same element in B — that is allowed.

Types of Functions — Know Every Category

1. One-One (Injective) Function

A function f : A → B is one-one if distinct elements in A always have distinct images in B. No two inputs give the same output.

Formally: a ≠ b ⟹ f(a) ≠ f(b)

Horizontal line test: If any horizontal line cuts the graph more than once, the function is NOT one-one. This is the fastest graphical check.

2. Onto (Surjective) Function

A function f : A → B is onto if every element in B has at least one pre-image in A. The range equals the co-domain: f(A) = B. No element in B is "uncovered."

3. Bijective (One-One and Onto) Function

A function that is both injective and surjective. Bijective functions create a perfect one-to-one pairing between A and B. Only bijective functions have a valid inverse.

4. Into Function

At least one element in B is NOT the image of any element in A. Range ≠ Co-domain. Most real-world mathematical functions are into functions.

5. Many-One Function

Multiple elements in the domain map to the same element in the co-domain. Example: f(x) = x² maps both 2 and −2 to 4. Many-one functions are not one-one.

\Note: A full inverse function $f^{-1} : B \to A$ exists only when $f$ is bijective. If $f$ is injective but not onto, an inverse exists from the range to the domain.*

Graph of a Function — Visual Understanding

The graph of a function is the set of all ordered pairs (x, f(x)). Graphing reveals function properties immediately.

Vertical line test: A curve represents a function if and only if no vertical line intersects it more than once. This is the fundamental graphical definition of a function.

Key graphs to know:

• Linear f(x) = mx + c with m ≠ 0 is one-one over R.
• Quadratic f(x) = x² → U-shaped parabola; many-one over all reals
• Absolute value f(x) = |x| → V-shaped; many-one over all reals
• Constant f(x) = c → Horizontal line; many-one (every input maps to same output)
• Identity f(x) = x → Diagonal line; always bijective

Composite Functions

If f : A → B and g : B → C, the composite function gof : A → C is:

Apply f first, then g. Critical points:

• gof ≠ fog in general — order matters
• If both f and g are one-one, then gof is one-one
• If both f and g are onto, then gof is onto
• Composite functions model multi-step processes: think of f as applying a formula and g as applying a further transformation.

Inverse Functions — The Bijection Requirement

A full inverse function f⁻¹ : B → A exists only when f is bijective. If f is injective but not onto, an inverse exists from the range to the domain. If f maps x to y, then f⁻¹ maps y back to x.

Finding the inverse: Set y = f(x), solve for x in terms of y, then swap x and y to get f⁻¹(x).

Example: f(x) = 2x + 3

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

Key properties:

• (f⁻¹)⁻¹ = f (inverse of inverse returns original)
• (gof)⁻¹ = f⁻¹og⁻¹ (order reverses for composite inverses)
• The graph of f⁻¹ is the reflection of f's graph across the line y = x

Special Functions Worth Memorising

These appear frequently in both board exams and entrance tests:

• Identity Function f(x) = x: Every input maps to itself. Always bijective. Domain and range are identical.
• Constant Function f(x) = c: Every input gives the same output c. Not one-one (unless domain is a singleton). Range = {c}.
• Modulus Function f(x) = |x|: Output is always non-negative. Not one-one over all reals; one-one over [0, ∞).
• Signum Function: Returns −1 for negative x, 0 for x = 0, and +1 for positive x.
• Greatest Integer Function ⌊x⌋: Returns the largest integer ≤ x. Also called the floor function. Discontinuous at every integer.

Real-World Connections for Commerce Students

Functions directly model core commerce concepts:

• Cost function C(x): Total cost of producing x units — a function mapping quantity to cost
• Revenue function R(x): Income from selling x units
• Profit function P(x) = R(x) − C(x): A composite of revenue and cost

Understanding functions as relationships between inputs and outputs is foundational to the mathematical economics you'll study in Class 12 and beyond.

Common Mistakes to Avoid

1. Confusing range with co-domain. Co-domain is the target set B; range is the actual output set. Range ⊆ Co-domain always.
2. Trying to invert a non-bijective function. If f is not one-one and onto, it has no inverse over the full domain.
3. Misapplying composite function order. gof means: apply f first, then g.
4. Forgetting the vertical line test. Every x-value can have only one y-value.

Related Posts

• See also: Sets and Relations in Class 11 Commerce Maths – Complete Guide
• Related: Differentiation in Class 11 Commerce – Derivative Rules and Applications
• Explore: CUET Mathematics Preparation Strategy for Commerce Students

Summary & Study Action Plan

Functions form the bridge between basic algebra and calculus — and they appear in almost every major entrance exam for commerce students. Mastering the classification of function types gives you faster, more confident answers in multiple-choice sections.

📌 Practice identifying each function type from its graph and its algebraic rule. Five problems per type, two days of focused practice — this chapter will feel second nature before your next mock test.

Frequently Asked Questions (FAQ)

Q1: What is the difference between a relation and a function?
Every function is a relation, but not every relation is a function. A function requires each element of the domain to have exactly one image in the co-domain.

Q2: Can a function be both one-one and onto?
Yes. Such a function is called bijective. Only bijective functions have a valid inverse.

Q3: What is the domain and range of f(x) = |x|?
Domain is all real numbers (−∞, ∞). Range is all non-negative real numbers [0, ∞).

Q4: How do you verify a function is onto?
Check that every element in the co-domain B can be expressed as f(a) for some a ∈ A. Equivalently, confirm range = co-domain.

Q5: Is the inverse of f(x) = x² defined over all real numbers?
No. Since f(x) = x² is many-one over all reals, its inverse is only defined when the domain is restricted to [0, ∞).

Q6: Why are functions especially important for commerce students?
Functions model cost-revenue-profit relationships that appear directly in economics, business mathematics, and quantitative aptitude sections of entrance exams.