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Maps To Symbol — Definition, Formula & Examples

The maps to symbol (↦) shows what output a function assigns to a specific input. For example, x↦x2x \mapsto x^2 means each value xx is sent to its square.

The symbol ↦\mapsto (read "maps to") defines the rule of a function by indicating the image of an element under that function. Given a function f ⁣:A→Bf\colon A \to B, the notation x↦f(x)x \mapsto f(x) specifies that the element x∈Ax \in A is assigned the element f(x)∈Bf(x) \in B.

How It Works

The maps to symbol appears inside a function definition to describe what happens to each input. You write the input variable on the left of ↦\mapsto and the output expression on the right. For instance, f ⁣:R→R,  x↦3x+1f\colon \mathbb{R} \to \mathbb{R},\; x \mapsto 3x + 1 defines a function from the reals to the reals that sends every xx to 3x+13x+1. Notice that the arrow →\to connects the domain to the codomain (set to set), while ↦\mapsto connects an element to its image (value to value). These two arrows often appear together in a single function definition but play different roles.

Worked Example

Problem: Define the function that squares a number and adds 5, using maps to notation, then evaluate it at x=3x = 3.
Write the function rule: Use the maps to symbol to state the rule concisely.
f ⁣:R→R,  x↦x2+5f\colon \mathbb{R} \to \mathbb{R},\; x \mapsto x^2 + 5
Evaluate at x = 3: Replace xx with 3 in the expression on the right side of ↦\mapsto.
3↦32+5=143 \mapsto 3^2 + 5 = 14
Answer: The function sends 3 to 14, written 3↦143 \mapsto 14.

Why It Matters

The maps to symbol is standard notation in college-level algebra, analysis, and abstract algebra courses. It lets you define functions without naming them (anonymous or lambda-style definitions), which is especially useful in linear algebra when describing transformations like v↦Av\mathbf{v} \mapsto A\mathbf{v}.

Common Mistakes

Mistake: Confusing the arrow →\to with the maps to symbol ↦\mapsto.
Correction: Use →\to between sets (e.g., f ⁣:R→Rf\colon \mathbb{R} \to \mathbb{R}) and ↦\mapsto between elements (e.g., x↦x2x \mapsto x^2). They serve different purposes in the same definition.

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