# Injective Function — How to Prove It, and Examples

## What Is An Injective Function?

An **injective function** f:A→B is one where **distinct inputs map to distinct outputs.** Said the way it's usually proved: if f(a)=f(b), then a=b. The everyday name is a **one-one function** (or "one-to-one").

Two equivalent ways to say the same thing:

- **Forward form:** a≠b⟹f(a)≠f(b) (different inputs can't collide).
- **Contrapositive form:** f(a)=f(b)⟹a=b (a shared output forces the inputs to be equal).

The contrapositive form is the one you actually use in proofs, because it gives you an equation to solve.

## How Do You Prove A Function Is Injective?

A frequent search is _how do I prove a function is one-one?_ There are two standard routes — one algebraic, one graphical.

**The algebraic method (works for any function with a formula):**
1. Assume f(a)=f(b).
2. Write out the equation and simplify.
3. If every legal simplification forces a=b, the function is injective.

**The horizontal line test (works when you have a graph):** A function is injective exactly when **no horizontal line crosses its graph more than once.** If any horizontal line hits the curve twice, two different inputs share that output, and one-one fails.

## What Is The Difference Between Injective And Surjective?

- **Injective (one-one):** no output is shared. Distinct inputs give distinct outputs. Some outputs may go unused.
- **Surjective (onto):** every output is used. No element of the codomain is left out. Sharing is allowed.

They are independent — a function can be one without the other. For example, f(x)=e^x on R→R is injective (no two inputs share a value) but not surjective (negatives are never outputs). A function that is **both** is a bijective function, and only those have inverses.

## What Is The Number Of Injective Functions Between Two Finite Sets?

If A has m elements and B has n elements with n≥m, the number of injective functions from A to B is the falling factorial:

n×(n−1)×(n−2)×⋯×(n−m+1)=n!(n−m)!

If m>n, the count is **0** — there aren't enough distinct outputs to go around, a direct consequence of the pigeonhole principle.

## Examples Of Injective Functions

### Example 1

**Decide whether f: {1, 2, 3} → {p, q, r, s} given by f(1)=p, f(2)=q, f(3)=r is injective.**

No two inputs share an output, so the function is **injective**.

### Example 2

**Decide whether f: R → R, f(x)=x^2, is injective.**

Test the definition. Suppose f(a)=f(b), this gives a^2=b^2 which allows a=b or a=−b. Thus, the function is **not injective**.

### Example 3

**Prove that f: R → R, f(x)=5x−7, is injective.**

Assume f(a)=f(b), so 5a−7=5b−7, leading to a=b. Thus, f is **injective**.

### Example 4

**Decide whether f: R → R, f(x)=x^3, is injective.**

Assume f(a)=f(b), we find that a=b. Hence, f is **injective**.

### Example 5

**Decide whether f: R → R, f(x)=1/(x+1), is injective.**

Assume f(a)=f(b). Cross-multiplying leads to a=b, so f is **injective** on its domain.

### Example 6

**Count the injective functions from a 2-element set to a 4-element set.**

The calculation is 4! / 2! = 12. There are **12** injective functions.

## Why One-One Matters: "Injectivity Is What Makes Reversal Possible"

Injectivity guarantees that a process can be undone without ambiguity. It is half of invertibility, which means a function can be reversed only if each output traces back to exactly one input.

## Where Injectivity Slips Away

Common mistakes include forgetting about negative roots, confusing injectivity with surjectivity, and misinterpreting inverse notation.

## Conclusion

- An **injective function** sends distinct inputs to distinct outputs.
- Prove it algebraically or graphically.
- Injective and surjective are independent properties.
- The number of injective functions can be calculated based on the sizes of sets.
- Injectivity is crucial for defining invertibility.