One-to-One Mapping

Functions
definition

Also known as: injective function, 1-to-1, one-to-one function, one-to-one

Grade 6-8

View on concept map

A one-to-one (injective) function maps every distinct input to a distinct output — no two different inputs produce the same output. One-to-one functions are precisely those that have inverse functions — this is why the horizontal line test (one-to-one check) is the prerequisite for finding an inverse.

Definition

A one-to-one (injective) function maps every distinct input to a distinct output — no two different inputs produce the same output.

💡 Intuition

No two inputs share the same output—like social security numbers.

🎯 Core Idea

One-to-one (injective) functions have unique outputs for each input.

Example

f(x) = 2x is one-to-one. f(x) = x^2 is NOT (both 2 and -2 give 4).

Formula

f(a) = f(b) \implies a = b

Notation

f(a) = f(b) \implies a = b is the algebraic test for one-to-one (injective). Graphically: horizontal line test.

🌟 Why It Matters

One-to-one functions are precisely those that have inverse functions — this is why the horizontal line test (one-to-one check) is the prerequisite for finding an inverse.

💭 Hint When Stuck

Try the horizontal line test: slide a horizontal line up and down the graph. If it ever crosses more than once, the function is not one-to-one.

Formal View

f\colon X \to Y is injective \iff \forall\,a, b \in X: f(a) = f(b) \implies a = b

🚧 Common Stuck Point

Test: horizontal line hits graph at most once \to one-to-one.

⚠️ Common Mistakes

  • Confusing one-to-one with onto — one-to-one means different inputs give different outputs; onto means every possible output is hit
  • Thinking f(x) = x^2 is one-to-one — it fails because f(2) = f(-2) = 4; two different inputs give the same output
  • Forgetting the horizontal line test — a function is one-to-one if and only if every horizontal line crosses the graph at most once

Frequently Asked Questions

What is One-to-One Mapping in Math?

A one-to-one (injective) function maps every distinct input to a distinct output — no two different inputs produce the same output.

Why is One-to-One Mapping important?

One-to-one functions are precisely those that have inverse functions — this is why the horizontal line test (one-to-one check) is the prerequisite for finding an inverse.

What do students usually get wrong about One-to-One Mapping?

Test: horizontal line hits graph at most once \to one-to-one.

What should I learn before One-to-One Mapping?

Before studying One-to-One Mapping, you should understand: function definition.

Prerequisites

function definition

How One-to-One Mapping Connects to Other Ideas

To understand one-to-one mapping, you should first be comfortable with function definition. Once you have a solid grasp of one-to-one mapping, you can move on to inverse function and horizontal line test.

← Back to Explorer