One-to-One Mapping Formula

The Formula

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

When to use: No two inputs share the same output—like social security numbers.

Quick Example

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

Notation

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

What This Formula Means

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

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

Formal View

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

Worked Examples

Example 1

easy
Determine whether f(x) = 2x + 3 is one-to-one by (a) the definition and (b) the horizontal line test.

Solution

  1. 1
    Definition: assume f(a)=f(b). Then 2a+3=2b+3 \Rightarrow 2a=2b \Rightarrow a=b. So f(a)=f(b) \Rightarrow a=b. f is one-to-one. ✓
  2. 2
    Horizontal line test: f(x)=2x+3 is a line with positive slope. Any horizontal line y=c intersects it at exactly one point (x = (c-3)/2).
  3. 3
    Both methods confirm f is one-to-one (injective).

Answer

f(x)=2x+3 is one-to-one
A one-to-one function has no two distinct inputs sharing an output: f(a)=f(b)\Rightarrow a=b. Linear functions with non-zero slope are always one-to-one because they are strictly monotone.

Example 2

medium
Show that g(x) = x^3 is one-to-one on \mathbb{R}, then find its inverse function.

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

Why This Formula 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.

Frequently Asked Questions

What is the One-to-One Mapping formula?

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

How do you use the One-to-One Mapping formula?

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

What do the symbols mean in the One-to-One Mapping formula?

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

Why is the One-to-One Mapping formula important in Math?

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 get wrong about One-to-One Mapping?

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

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

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

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