Discrete Mathematics · Quick Reference

Functions Cheat Sheet

Discrete Mathematics · Lesson 4/9

In one line: a function is a special type of relation where every input maps to exactly one output — one of the most fundamental building blocks across all of mathematics.

Key Ideas

1Function Definition. A relation from A to B where every element of A is paired with exactly one element of B.

2Injective (One-to-One). A function where different inputs always produce different outputs — no two inputs share the same output.

3Surjective (Onto). A function where every element of the output set (codomain) is actually reached by at least one input.

4Bijective. A function that is both injective and surjective — every output is hit exactly once, creating a perfect pairing between inputs and outputs.

5Composition of Functions. Applying one function to the result of another: (f composed with g)(x) = f(g(x)) — compute the inner function first.

Worked Examples

Is f(x) = x^2 injective on all real numbers?

No, f(x)=x^2 is not injective (fails at x=2 and x=-2)

Is f(x) = x + 3 injective on all real numbers?

Yes, f(x) = x+3 is injective

Let f(x) = x+1 and g(x) = 2x. Find (f composed with g)(3).

(f composed with g)(3) = 7

Formulas

(f composed with g)(x) = f(g(x))

Practice Yourself

Let f(x)=2x and g(x)=x+5. Find (f composed with g)(2).

f(g(2))=f(7)=14

Is f(x) = 3x injective?

Yes

Is f(x) = |x| injective on all real numbers?

No, f(2)=f(-2)=2