Category: Foundational Algebra & Set Mapping
In algebra, a function is a strict relation that assigns each input value from a set called the Domain to exactly one unique output value in a set called the Codomain (or Range). You can think of a function like a computer program or a machine: you feed it a raw parameter x, it executes a specific arithmetic rule, and it spits out a predictable output f(x).
Not every mathematical equation or graph qualifies as a function. The absolute rule of functions states that one input cannot lead to multiple different outputs. To test a graph visually, we utilize the Vertical Line Test: if you can draw a vertical line anywhere through a graph and intersect the curve more than once, the relation fails to be a function (like the equation of a circle).
When you feed the output of one function directly into another, you perform a Composition of Functions, written as (g ∘ f)(x) = g(f(x)). Let us prove mathematically that function compositions are generally non-commutative, meaning that the sequential order of operations fundamentally changes the final output.
This algebraic proof reveals why order of execution matters. This is a crucial concept for video game programming, where tracking whether you multiply a player's speed before or after applying a slowdown debuff completely changes the player's movement physics.
Now that you understand standard algebraic input mapping, you are ready to see how functions behave when inputs get infinitely close to a value. Check out our entry on Limits to Infinity and Mathematical Continuity!