One-to-One Mapping Formula

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

The Formula

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

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

Quick Example

f(x)=2xf(x) = 2x is one-to-one. f(x)=x2f(x) = x^2 is NOT (both 2 and −2-2 give 4).

Notation

f(a)=f(b)  ⟹  a=bf(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 ⁣:X→Yf\colon X \to Y is injective   ⟺  \iff ∀ a,b∈X:f(a)=f(b)  ⟹  a=b\forall\,a, b \in X: f(a) = f(b) \implies a = b

Worked Examples

Example 1

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

Answer

f(x)=2x+3f(x)=2x+3 is one-to-one

First step

1
Definition: assume f(a)=f(b)f(a)=f(b). Then 2a+3=2b+3⇒2a=2b⇒a=b2a+3=2b+3 \Rightarrow 2a=2b \Rightarrow a=b. So f(a)=f(b)⇒a=bf(a)=f(b) \Rightarrow a=b. ff is one-to-one. ✓

Full solution

  1. 2
    Horizontal line test: f(x)=2x+3f(x)=2x+3 is a line with positive slope. Any horizontal line y=cy=c intersects it at exactly one point (x=(c−3)/2)(x = (c-3)/2).
  2. 3
    Both methods confirm ff is one-to-one (injective).
A one-to-one function has no two distinct inputs sharing an output: f(a)=f(b)⇒a=bf(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)=x3g(x) = x^3 is one-to-one on R\mathbb{R}, then find its inverse function.

Example 3

medium
Show that f(x)=5−2xf(x) = 5 - 2x is one-to-one and find its inverse.

Common Mistakes

  • Confusing one-to-one with being a function - a function needs one output per input; one-to-one needs one input per output.
  • Using the vertical line test for one-to-one - one-to-one is checked by the horizontal line test.
  • Calling a function one-to-one when an output repeats - any repeated output value breaks it.

Why This Formula Matters

One-to-one is the exact property a function needs to be reversible — without it, an inverse would have to send one output back to two inputs. It underlies encryption, decoding, and every 'solve for the input' problem. Recognizing it by "Does every output value come from at most one input?" — rather than by familiar numbers — is what lets a student tell it apart from many-to-one mapping and function (in general) and onto (surjective) in a mixed problem set.

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)  ⟹  a=bf(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 is the exact property a function needs to be reversible — without it, an inverse would have to send one output back to two inputs. It underlies encryption, decoding, and every 'solve for the input' problem. Recognizing it by "Does every output value come from at most one input?" — rather than by familiar numbers — is what lets a student tell it apart from many-to-one mapping and function (in general) and onto (surjective) in a mixed problem set.

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

The procedure for one-to-one mapping is the easy part; the trap is confusing one-to-one with being a function. Asking "Does every output value come from at most one input?" first is what keeps a correct-looking calculation from being attached to the wrong concept.

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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