One-One and Onto Functions: Injective, Surjective Guide

A function is one-one (injective) when different inputs always give different outputs: no two arrows land on the same target. It is onto (surjective) when every element of the codomain is hit by at least one input. A function that is both is called bijective.

These properties matter because they decide what a function can do. Only a bijective function can be undone by an inverse, and counting arguments in combinatorics constantly rely on matching sets through bijections.

Key Concepts

• One-one (injective): f(a) = f(b) implies a = b; distinct inputs give distinct outputs.
• Onto (surjective): for every y in the codomain there is at least one x with f(x) = y; the range equals the codomain.
• Bijective: both one-one and onto; sets up a perfect one-to-one pairing between domain and codomain.
• Horizontal line test: a graph represents a one-one function exactly when every horizontal line meets it at most once.

How It Works

1. One-one test: assume f(a) = f(b) and simplify. If you can force a = b, the function is one-one; if you find a counterexample with a not equal to b, it is not.
2. Onto test: take an arbitrary y in the codomain and try to solve y = f(x) for x. If a solution in the domain always exists, the function is onto.
3. Combine: pass both tests and the function is bijective, which guarantees an inverse exists.

Worked Example

Test f: R to R given by f(x) = 3x minus 5.

1. One-one: suppose 3a minus 5 = 3b minus 5. Adding 5 to both sides gives 3a = 3b, dividing by 3 gives a = b. One-one confirmed.
2. Onto: for any real y, solve y = 3x minus 5 to get x = (y + 5)/3, which is a real number. Every y is hit, so onto confirmed.
3. Numeric check: f(4) = 12 minus 5 = 7, and the input producing 10 is x = (10 + 5)/3 = 5, since f(5) = 15 minus 5 = 10.
4. f is bijective, so it has an inverse: f inverse of x = (x + 5)/3.

Common Misconceptions

• "One-one means strictly increasing." Not necessarily: f(x) = minus x is strictly decreasing yet one-one, since different inputs still give different outputs.
• "Onto means it hits everything in the domain." Onto is about the codomain: every declared output must be reached, whatever the domain looks like.
• "f(x) = x squared is bijective." On R to R it is neither: f(2) = f(minus 2) breaks one-one, and negative numbers are never hit, breaking onto.

Real-World Applications

• Cryptography: an encryption function must be bijective so every coded message decodes to exactly one original.
• Numbering systems: assigning each student a unique roll number is a one-one function from students to numbers.
• Unit conversion: Celsius to Fahrenheit is bijective, which is why the conversion works perfectly in both directions.

One-one means no two inputs share an output; onto means no codomain element is left out; bijective means both. Test with algebra — f(a) = f(b) implies a = b, and y = f(x) solvable for all y — and these properties become routine to verify.