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# Onto Function — Definition, Formula, Graph, and Examples

## TL;DR

An onto function (also called a surjective function) is one where every element of the codomain is the output of at least one input — nothing in the target set is left unreached. This article gives the formal definition, the test (range equals codomain), the formula for the number of onto functions, the onto-vs-into contrast, and six worked examples.

## What Is An Onto Function?

An **onto function** f:A→B is one where **every element of the codomain B is the image of at least one element of the domain A.** Written formally: for every y∈B, there exists at least one x∈A such that f(x)=y. The textbook synonym is a **surjective function** — the map sends A _onto_ all of B. The companion article on the [surjective function](/content/math/algebra/surjective-function/index.html) walks the same idea through a formal onto-proof if you want the proof-heavy version.

There is no uniqueness condition. Two different inputs may land on the same output; the only thing forbidden is leaving an output unreached. The one-line working test is this:

**A function is onto exactly when its range equals its codomain.**

- **Codomain** — the target set you _declare_ when you write f:A→B. It is a choice.
- **Range** — the set of outputs the function _actually_ produces.

The range always sits inside the codomain. Onto means the two are the same set, with nothing left over.

## How Do You Check If A Function Is Onto?

A common search is simply _how do you know if a function is onto?_ The procedure runs backwards, starting from an arbitrary output:

1. Take an arbitrary element y in the **codomain** B.
2. Solve f(x)=y for x.
3. Confirm that solution x actually lies in the **domain** A.

If you can always produce such an x, every output is reachable and the function is onto. On a graph, the parallel idea is the **horizontal line test for onto**: a function is onto its codomain when **every horizontal line drawn through the codomain meets the graph at least once.**

## What Is The Difference Between An Onto Function And An Into Function?

The two terms are complements:

- **Onto (surjective):** every element of B is reached. Range === codomain.
- **Into:** at least one element of B is _not_ reached by any input. Range is a **proper subset** of the codomain.

Every function is either onto or into — there is no third option once the codomain is fixed. The classic example is f(x)=x² written as f:R→R. No negative number is ever an output, so −4 has no preimage. The range is [0,∞), a proper subset of R, which makes the function **into**. The [into function](/content/math/algebra/into-function/index.html) article works this contrast from the other side. Rewrite the codomain as f:R→[0,∞) and the same rule becomes onto.

## What Is The Formula For The Number Of Onto Functions?

When A has m elements and B has n elements, the number of onto functions from A to B is counted by inclusion-exclusion:

Number of onto functions=∑k=0n(−1)k(nk)(n−k)m

A few consequences worth carrying:

- If m<n, there are **0** onto functions — there aren't enough inputs to cover every output.
- If m=n, the count is n!, and every onto function in this case is also one-one (a [bijective function](/content/math/algebra/bijective-function/index.html)).
- The formula relies on the [binomial theorem](/content/math/algebra/binomial-theorem/index.html) coefficient (nk), which counts how many ways k outputs get left out.

Each symbol means: m is the size of the domain, n the size of the codomain, (nk) the number of ways to choose k codomain elements to exclude, and (n−k)m the number of functions mapping into the remaining n−k outputs.

## Examples Of Onto Functions

### Example 1

**Decide whether f:{1,2,3}→{a,b} given by f(1)=a, f(2)=b, f(3)=b is onto.**

Check each codomain element for an incoming arrow.

a is reached by 1.

b is reached by 2 and 3.

Both elements are reached, so the function is **onto**. The repeated output (b twice) is allowed.

### Example 2

**Decide whether f:R→R, f(x)=x², is onto.**

The tempting first move is to say "every output has a square root, so it's onto."

Test it on a specific output. Pick y=−4 in the codomain R.

Solve x²=−4.

There is no real x whose square is −4.

So −4 is never produced. The function is **not onto**; its range is only [0,∞). Fix the codomain to [0,∞) and it becomes onto.

### Example 3

**Show that f:R→R, f(x)=2x−1, is onto.**

Take an arbitrary y∈R.

Solve 2x−1=y.

x=y+1/2.

This x is a real number for every real y, so it lies in the domain.

Every output is reachable, so f is **onto**.

### Example 4

**Decide whether f:R→R, f(x)=x³, is onto.**

Take an arbitrary y∈R.

Solve x³=y.

x=y³.

Every real number has a real cube root, so x is always in the domain.

The function is **onto** (and, since cubing is also one-one, it is bijective).

### Example 5

**Count the onto functions from a 3-element set to a 2-element set.**

Here m=3, n=2.

∑k=02(−1)k(2k)(2−k)³=8−2+0=6.

There are **6** onto functions.

### Example 6

**A factory has 4 machines, each routed to exactly one of 3 output bins. How many routings use every bin at least once?**

This is "onto functions from a 4-element set to a 3-element set," so m=4, n=3.

∑k=03(−1)k(3k)(3−k)⁴=36.

So **36** routings leave no bin empty.

## Why The Codomain Matters: "Onto Is A Claim About Coverage"

Onto exists because mathematics needed a precise way to say _the map covers the whole target_. That single idea decides whether a function can be reversed at all.

- **It is the gatekeeper for inverses.** A function can be undone only if it is onto (and one-one). If outputs are missing, there is nothing to map them back from. This is the foundation under every [inverse function](/content/math/algebra/inverse-functions/index.html).

- **It powers counting arguments.** Cryptographic key assignment, hashing into a fixed number of buckets, and scheduling problems all ask "does every slot get used?" — a surjectivity question.

- **It underwrites the pigeonhole reasoning** behind why a finite map can be onto only when the domain is at least as large as the codomain.

The destination this opens up is **cardinality**: comparing the sizes of infinite sets. The mathematician [Georg Cantor](https://mathshistory.st-andrews.ac.uk/Biographies/Cantor/) used onto and one-one maps in the 1870s to prove that some infinities are genuinely larger than others.

## Tripping Points To Avoid

### Mistake 1: Confusing range with codomain

**Where it slips in:** any question where the function's formula produces fewer outputs than the declared target set.

**Don't do this:** declare f:R→R, f(x)=eˣ onto because "exponentials grow without bound."

**The correct way:** check the _full_ codomain. eˣ is always positive, so 0 and every negative number are never outputs. Range is (0,∞), not R, so it is into, not onto.

### Mistake 2: Treating a repeated output as a failure

**Where it slips in:** finite mapping diagrams where two inputs point at the same output.

**Don't do this:** call f(1)=a, f(2)=a, f(3)=b (codomain a,b) not-onto because "1 and 2 collide."

**The correct way:** collisions are irrelevant to onto-ness. Ask only whether every output is reached. Both a and b are reached, so it is onto.

### Mistake 3: Forgetting that onto depends on the codomain

**Where it slips in:** problems that hand you a formula without naming the codomain.

**Don't do this:** answer "onto" or "not onto" before the target set is fixed.

**The correct way:** treat the codomain as part of the question. f(x)=x² is _not onto_ as R→R but _is onto_ as R→[0,∞). Same formula, opposite answer.

## Conclusion

- An **onto function** reaches every element of its codomain — range equals codomain.
- Onto and into are complements: into leaves at least one output unreached.
- To check ontoness, solve f(x)=y for an arbitrary y and confirm x stays in the domain.
- The number of onto functions from an m-set to an n-set is ∑k=0n(−1)k(nk)(n−k)m.
- Ontoness depends on the declared codomain, and it is half of what a function needs to be invertible.