A function is like a predictable machine. You put something in, and it gives you exactly one thing back. Every single time. It's a rule that connects an input to a single, specific output.

This rule of 'one unique output' is what makes functions so reliable and important in mathematics. They establish a clear, unambiguous relationship between two sets of values.

In the diagram above, every number in Set X is paired with exactly one letter in Set Y. Notice that input 1 always goes to output B, never to anything else. That's what makes this a function.

But not all relationships work this way. A relationship that pairs an input with more than one output is just a 'relation', not a function.

This second example is not a function because the input '2' is linked to two different outputs, 'A' and 'C'. It breaks our 'one unique output' rule.

Functions have specific names for their sets of inputs and possible outputs. These are the domain and the codomain.

Let's look at our function example again, but with these new terms.

The domain is the set {1, 2, 3}, and the codomain is the set {A, B, C, D}. It's perfectly fine for an item in the codomain (like 'D') to not be an output for any input. The codomain is just the set of potential outputs.

Writing out "the function that takes an input and doubles it" is clumsy. Mathematicians use a shorthand called function notation to describe these rules concisely.

We usually use letters like $f$, $g$, or $h$ to name functions. Then we write $f(x)$ to represent the output of the function $f$ when the input is $x$. You would read "$f(x)$" as "f of x".

Let's define a function that doubles the input number. We can write this rule as:

This notation gives us a template. To find the output for any input, we just replace $x$ with that input's value.

If we want to evaluate our function for an input of 3, we write $f(3)$.

The input is 3, and the output is 6. What about an input of -5?

Let's try one more example. Consider a function $g$ defined by:

This rule says to take the input, square it, and then add 1. To find $g(4)$, we substitute 4 for $x$:

Function notation is a powerful and efficient way to define the relationship between inputs and outputs.

Functions are a fundamental building block in mathematics, providing a clear way to model relationships between quantities.