Bijective and Invertible Functions — Class 12 Maths Chapter 1 Complete Guide

Two concepts that consistently appear in CBSE board questions and competitive exams are bijective functions and invertible functions. They are deeply connected: a function has an inverse if and only if it is bijective. Once you understand this relationship, an entire class of exam questions becomes straightforward.

This guide covers both concepts with precise definitions, worked examples from the NCERT textbook, and the exam strategies you need.

What Is a Bijective Function?

A function f : X → Y is called bijective (or a one-one correspondence) if it is simultaneously:

  • One-one (injective): distinct inputs give distinct outputs
  • Onto (surjective): every element of the co-domain has a pre-image in the domain

A bijective function creates a perfect one-to-one pairing between every element of X and every element of Y — no elements are left out, and none are double-counted.

Why Bijectivity Matters

Bijectivity is not just a theoretical nicety. It is the exact condition under which a function can be reversed — which brings us to invertibility.

Classic Examples

Bijective: f : R → R, f(x) = 2x
One-one since f(x₁) = f(x₂) implies x₁ = x₂. Onto since for any y ∈ R, x = y/2 satisfies f(x) = y. ✓

Bijective: f : N → N, f(x) = x + 1 if x is odd, f(x) = x − 1 if x is even
Each odd number is swapped with the even number above it. Both injectivity and surjectivity hold. ✓

Not bijective: f : R → R, f(x) = x²
Neither one-one (f(−1) = f(1)) nor onto (−2 has no real square root). ✗

What Is an Invertible Function?

A function f : X → Y is called invertible if there exists a function g : Y → X such that:

g ∘ f = I_X and f ∘ g = I_Y

Here I_X and I_Y are the identity functions on X and Y respectively. The function g is called the inverse of f and is written f⁻¹.

The Golden Rule

A function is invertible if and only if it is bijective.

This means:

  • To prove f is invertible, prove f is one-one and onto.
  • To prove f is not invertible, show it is either not one-one or not onto (or both).
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How to Find the Inverse of a Function

Finding f⁻¹ is a two-step process:

Step 1: Express the equation y = f(x) and solve for x in terms of y.
Step 2: Define g(y) using this expression, then verify that g ∘ f = I_X and f ∘ g = I_Y.

Worked Example (NCERT)

Let $f : \mathbb{N} \to Y$, $f(x) = 4x + 3$, where $Y = \{y \in \mathbb{N} : y = 4x + 3 \text{ for some } x \in \mathbb{N}\}$. Prove $f$ is invertible and find its inverse.

WORKED EXAMPLE · Step-by-Step Solution: Invertibility and Finding f⁻¹(y)
Given f : N → Y with f(x) = 4x + 3, where Y is the range of f. Prove that f is invertible and determine the inverse function f⁻¹.
Walk through injectivity, surjectivity, and solving for x in terms of y:
Step 1: Prove Injective (One-One)
f(x_1) = f(x_2) \implies 4x_1 + 3 = 4x_2 + 3 \implies 4x_1 = 4x_2 \implies x_1 = x_2
Since f(x₁) = f(x₂) implies x₁ = x₂, the function is strictly one-one.
Step 2: Prove Surjective (Onto)
y∈Y  ⟹  y=4x+3 for some x∈N  ⟹  x=y−34∈Ny \in Y \implies y = 4x + 3 \text{ for some } x \in \mathbb{N} \implies x = \frac{y - 3}{4} \in \mathbb{N}
By definition of set Y, every element y is an image of some natural number x = (y - 3)/4. Hence f is onto.
Step 3: Define the Inverse Function g(y)
g:Y→N,g(y)=y−34g : Y \to \mathbb{N}, \quad g(y) = \frac{y - 3}{4}
Solving y = 4x + 3 for x gives the inverse formula g(y).
Step 4: Verify Identity Composition
(g∘f)(x)=g(4x+3)=(4x+3)−34=x=IN(x),(f∘g)(y)=4(y−34)+3=y=IY(y)(g \circ f)(x) = g(4x + 3) = \frac{(4x + 3) - 3}{4} = x = I_N(x), \quad (f \circ g)(y) = 4\left(\frac{y - 3}{4}\right) + 3 = y = I_Y(y)
Both compositions yield identity functions, rigorously confirming f⁻¹(y) = (y - 3)/4.
Solution Complete!
All 4 steps revealed and verified.

Let $f : \\mathbb{N} \\to Y$, $f(x) = 4x + 3$, where $Y = \\{y \\in \\mathbb{N} : y = 4x + 3 \\text{ for some } x \\in \\mathbb{N}\\}$. Prove $f$ is invertible and find its inverse.

Let f : N → Y, f(x) = 4x + 3, where Y = {y ∈ N : y = 4x + 3 for some x ∈ N}.

Showing f is bijective:
One-one: f(x₁) = f(x₂) ⟹ 4x₁ + 3 = 4x₂ + 3 ⟹ x₁ = x₂. ✓
Onto: Every element of Y is, by definition, of the form 4x + 3. ✓

Finding the inverse:
From y = 4x + 3, we get x = (y − 3)/4. Define g : Y → N by g(y) = (y − 3)/4.

Verification:
g(f(x)) = g(4x + 3) = (4x + 3 − 3)/4 = x = I_N(x). ✓
f(g(y)) = f((y − 3)/4) = 4·(y − 3)/4 + 3 = y = I_Y(y). ✓

So f⁻¹(y) = (y − 3)/4.

Interactive Simulation: Exploring Invertibility & Bijective Mapping

Interactive What-If: Testing Invertibility & Bijective Mapping

Adjust slope 'a' of function f(x) = ax + b. A non-zero slope guarantees bijection and an inverse, while slope a = 0 makes f(x) constant (many-one, not onto) and destroys the inverse.

Function f(x) = ax + b

Slope a4
Intercept b3
Input x2
Output f(x)11
Inverse f⁻¹(x) -0.2
Inverse Exists? Yes

Function g(x) = cx + d

Slope c1
Intercept d-3
Input x2
Output g(x)-1

Compositions

(f ∘ g)(x) = f(g(x)) -1
(g ∘ f)(x) = g(f(x)) 8
Is Commutative? (f ∘ g = g ∘ f) No ✗
Syllabus Checkpoints & Exam Watch-Outs
  • A function is invertible iff it is bijective. A zero slope makes it constant and non-invertible.

Bijective Functions on Finite Sets

For finite sets, there is an important shortcut: on a finite set X, a function f : X → X is one-one if and only if it is onto. This means every bijection from a finite set to itself is simply a permutation of that set.

This also explains why the number of bijective functions from {1, 2, 3} to itself equals 3! = 6 — the number of permutations of three elements.

Composition and Bijectivity

If f : A → B and g : B → C are both bijective, then their composition g ∘ f : A → C is also bijective. Moreover, (g ∘ f)⁻¹ = f⁻¹ ∘ g⁻¹. This mirrors the reversal rule familiar from matrix inverses.

🎯 Master Bijective and Invertible Functions

Work through exam-style questions on bijections and inverses with instant, step-by-step AI explanations on ChampionsPrep.

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Test Your Knowledge

Q1.What is the necessary and sufficient condition for a function f : X → Y to be invertible?
Q2.Why is the function f : ℝ → ℝ defined by f(x) = x² not invertible?
Q3.How many bijective functions exist from a finite set of 4 elements to itself?

Frequently Asked Questions

What is the difference between a bijective function and an invertible function? +

They are the same thing. A function is invertible if and only if it is bijective — both one-one and onto.

How do you check if a function is bijective? +

Check that it is one-one (f(x₁) = f(x₂) implies x₁ = x₂) and onto (every element of the co-domain has a pre-image). Both conditions must hold.

Can f : R → R, f(x) = x² be invertible? +

No. It is neither one-one nor onto, so it is not bijective and therefore not invertible.

What does it mean for gof = I_X? +

It means g undoes whatever f does — if f maps x to f(x), then g maps f(x) back to x. This establishes g as the inverse of f.

How many bijective functions exist from {1, 2, 3} to itself? +

Six — corresponding to the 3! = 6 permutations of three elements, since a bijection from a finite set to itself is a permutation.

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